Algebra and Discrete Mathematics

A Festschrift in Honor of Anthony Gaglione

Algebra and Discrete Mathematics ISSN: 1793-5873

Managing Editor: RUdiger Gobel (University Duisburg-Essen, Germany)

Editorial Board: Elisabeth Bouscaren, Manfred Droste, Katsuya Eda, Emmanuel Dror Farjoun, Angus MacIntyre, H.Dugald Macpherson, Jose Antonio de la Pefia, Luigi Salce, Mark Sapir, Lutz Strlingmann, Simon Thomas

The series ADM focuses on recent developments in all branches of algebra and topics closely connected. In particular, it emphasizes combinatorics, set theoretical methods, model theory and interplay between various fields, and their influence on algebra and more general discrete structures. The publications ofthis series are of special interest to researchers, post-doctorals and graduate students. It is the intention of the editors to support fascinating new activities of research and to spread the new developments to the entire mathematical community.

Vol. I: Aspects of Infinite Groups: A Festschrift in Honor of Anthony Gaglione eds. Benjamin Fine, Gerhard Rosenberger & Dennis Spellman

Algebra and Discrete Mathematics

I" INIT~ G~O ~ A Festschrift in Honor of Anthony Gaglione

editors

Benjamin Fine Fairfield University, USA

Gerhard Rosenberger Universitat Dortmund, Germany

Dennis Spellman Temple University, USA

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Algebra and Discrete Mathematics - Vol. 1 ASPECTS OF INFINITE GROUPS A Festschrift in Honor of Anthony Gaglione

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ASPECTS OF INFINITE GROUPS

Editors: Benjamin Fine, Gerhard Rosenberger, Dennis Spellman

PREFACE

This volume consists of contributions by participants and speakers at the conference entitled Aspects of Infinite Groups held at Fairfield University in March 2007 in honor of Prof. Anthony Gagliones sixtieth birthday.

Prof. Gaglione of the United States Naval Academy at Annapolis has made important and varied contributions to several different areas of Infi­nite Group Theory and Combinatorial Group Theory. Along with Herman Waldinger he developed and studied many important topics within the commutator calculus. Along with Dennis Spellman he proved a magnifi­cent result tying together the concepts of residual freeness and universal freeness. This result was also done independently by Vladmir Remeslen­nikov and can be considered as one of the seminal steps in the final proof of the celebrated Tarski problems. The final proof of the Tarski problems was accomplished by O.Kharlampovich and A.Myasnikov and indepen­dently by Z.Sela. Finally Prof. Gaglione, along with G.Baumslag, B.Fine, A.Myasnikov, V.Remeslennikov and D.Spellman completely developed the theory of discriminating and sqaurelike groups. This important class of groups was introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov as an outgrowth of the development of algebraic geometry over groups.

The papers in this volume provide an interesting mix of results on modern infinite discrete group theory ranging from classical combinato­rial group theory to algebraic geometry over groups to noncommutative algebraic cryptography.

The main speakers were primarily people who worked closely with Prof. Gaglione. They were

Prof. Michael Anshel City University of New York, New York City, NY

Prof. Gilbert Baumslag City University of New York, New York City, NY

Prof. Benjamin Fine

v

vi

Fairfield University, Fairfield, CT Prof. Alexei Myasnikov

McGill University, Montreal, Canada Prof. Gerhard Rosenberger

TU Dortmund, Dortmund, Germany Prof. Dennis Spellman

Temple University, Philadelphia, Pennsylvania

Prof. Gaglione received his Ph.D. in Mathematics from Polytechnic Uni­versity in Brooklyn, New York in 1972. He has been a professor at the United States Naval Academy since 1977.

He is the author or coauthor of more than 70 journal articles represent­ing the areas of combinatorial group theory described earlier. He is also the author or coauthor of four books.

Benjamin Fine Gerhard Rosenberger Dennis Spellman

CONTENTS

Preface

Actions, Commutator Identities, and the Algebraic Eraser™ I. Anshel, M. Anshel and D. Goldfeld

Virtually Free-By-Cyclic One-Relator Groups: I G. Baumslag and D. Troeger

Some Cryptoprimitives for Noncommutative Algebraic Cryptography

G. Baumslag, Y. Bryukhov, B. Fine and G. Rosenberger

On the Derived Subgroups of the Free Nilpotent Groups of Finite Rank

R. D. Blyth, P. Moravec and R. F. Morse

A Recurrence Relation for the Number of Free Subgroups in Free Products of Cyclic Groups

T. Camps, M. Darfer and G. Rosenberger

The Baumslag-Solitar Groups: A Solution for the Isomorphism Problem

A. E. Clement

Unification Theorems in Algebraic Geometry E. Daniyarova, A. Myasnikov and V. Remeslennikov

Reflections on Commutative Transitivity B. Fine and G. Rosenberger

Groups Universally Equivalent to Free Burnside Groups of Prime Exponent and a Question of Philip Hall

A. Gaglione, S. Lipschutz and D. Spellman

Changing Generators in Nonfree Groups R. Goldstein

vii

v

1

9

26

45

54

75

80

112

131

149

viii

Matrix Completions Over Principal Ideal Rings 151 W. H. Gustafson, D. W. Robinson, R. B. Richter and W. P. Wardlaw

A Primer on Computational Group Homology and Cohomology 159 D. Joyner

Doubles of Residually Solvable Groups 192 D. K ahrobaei

An Application of a Group of Ol'shanskii to a Question of Fine et. aZ. 201

S. Lipschutz and D. Spellman

Quotient Isomorphism Invariants of a Finitely Generated Coxeter Group 212

M. Mihalik, J. Ratcliffe and S. Tschantz

Localization and I A-automorphisms of Finitely Generated, Metabelian, and Torsion-Free Nilpotent Groups 228

M. Zyman

ACTIONS, COMMUTATOR IDENTITIES, AND THE ALGEBRAIC ERASER™

Iris Anshel

31 Peter Lynas Ct. Tenafly, NJ 07670, USA

Michael Anshel

City College of New York, New York, NY 10031, USA

Dorian Goldfeld'

Columbia University, Department of Mathematics, New York, NY 10027

Dedicated to Tony Gaglione on his 60th birthday.

Abstract: An algebraic structure arising in the formulation of a lightweight key agreement protocol yieldsa new concept of commutator whose identities are quite traditional

1. Introduction

In [AAGL ] the authors introduced a key agreement protocol for public­key cryptography suitable for implementation on lightweight platforms,that is, those subject to severe cost and resource constraints. Careful examina­tion reveals a hidden notion of commutator possessing identities analagous to those formulated by P. Hall, W. Magnus, E. Witt. The question as to whether or not these eminent mathematicians of the twentieth century de­veloped the theory of commutators for cryptographic purposes was raised in [AG]. Our method is to formulate the necessary mathematical primitives as monoid (group) actions and then identify the objects of interest. We conclude by inviting the reader to explore a certain action in the context of

'The authors would like to thank SecureRF for its support of this research

2

nonhopfian groups. This paper is dedicated to Professor Anthony Gaglione on his 60th Birthday.

2. The Algebraic Eraser™ and its Key Agreement Protocol:

Let M, N denote monoids (or groups), and let

II: M -t N,

be a homomorphism. In addition, let S denote a monoid (or group) which acts on M on the left as a monoid of endomorphisms (or a group of auto­morphisms), i.e., S -t End(M), where we view End(M) as a monoid. The action of s E Son m E M, is denoted by 8 m , and the semi direct product of M and S, denoted M XI S, is the monoid (or group) constructed in the classical manner:

(ml,sr) (m2,s2) = (mI81m2, SIS2).

The Algebraic Eraser ™ is, in essence, a right action of M XI S in the direct product N x S (viewed as a set), together with some additional apparatus which allows for cryptographic applications. The action E is specified as follows:

E : (N x S) x (M XI S) -t N x S

is given by

In practice we often use the more compact notation,

That E is a right action follows from the elementary computation with the operation *: given (n, s) E N x Sand (ml' SI), (m2' S2) E M XI S then

((n, s) * (ml' sd) * (m2' S2) = (nII(8 m1 ), SSI) * (m2' S2)

= (nIIe(mf1m2)), SSI S2) = (n, s) * ((mr, S1) (m2' S2))'

From the point of view of effective computation, the identity above allows us to compute * iteratively, which will be useful for applications. In particular, observing that (1,1) * (m, s) = (II(m), s) for all (m, s) E M XI S, if one

3

expressed an element (m, s) as a product in M ><l S, then (II(m) , s) can be computed in stages.

Associated to the action E, and crucial to encryption applications, is the concept of E-commuting which is defined as follows: two sub mono ids (or subgroups) of A, B :::; M ><l S are said to E-commute provided that for all (a, sa) E A, (b, Sb) E B

(1) (II(a), Sa) * (b, Sb) = (II(b), Sb) * (a, sa).

With the above notations in place, the Algebmic Emser E is defined to be the compilation of the above data,

(M ><l S, N, II, E, A, B).

The term algebmic emser is a fitting description of our structure (M ><l

S,N,II,E,A,B) in that given (n,s) E N x S, and (ml,sd E M><l S, knowledge of

(n, s) * (ml' sd,

the element (ml' sd cannot generally be recovered since the action of the element son ml is not visible once the function II has been applied to sml i.e., the action of s on ml has been effectively emsed.

With the algebraic eraser E specified we are in a position to introduce an associated key agreement protocol. Referring to the protocol users as Alice and Bob, each user is assigned a submonoid of N, N A, N B :::; N respectively so that N A and N B commute. Furthermore we view the E­commuting submonoids A and B as assigned to Alice and Bob, respectively. With all these choices in place Alice and Bob can choose their respective

private keys as follows:

where na E N A , nb E NB,

(Wa, sa) = (al' Sa,)(a2, saJ··· (ak, Sak) E A :::; M ><l S,

and

(Wb, Sb) = (h, sb, )(b2, Sb 2 )'" (bR, Sbe) E B :::; M ><l S.

Having made these choices, Alice and Bob can then announce their respec­

tive public keys:

Apublic = (na, l)*(wa, sa) = (naII(wa), sa) = ((na, l)*(al, Sa, ))*(a2, Sa2 ))* ... E Nx S,

Bpublic = (nb, l)*(Wb, Sb) = (nbII(wb), Sb) = ((nb, id)*(bl, Sb, ))*(b2, Sb2 ))* ... E N X S.

4

With this done Alice and Bob are now each in a position to compute the shared secret: Alice computes

(na, 1) . (Bpublic * (Wa, sa)) = (nanb, 1) . ((II(wb), Sb) * (Wa, sa))

(here· denoted multiplication in N x S), Bob computes

(nb, 1) . (Apublic * (Wb, Sb)) = (nbna, 1) . ((II(wa), sa) * (Wb, Sb)).

The assumption that A and BE-commute and N A and N B commute, implies that the two expressions above, computed individually by Bob and Alice, coincide, and the exchanged key is given by

(nanbII(wgbwa), SbSa) = (nbnaII(w~awb), SaSb)

Specific and detailed implementations of the above protocol can be found in [AAGL]. It should ne noted that use of commuting monoids and the E-commuting structures A, B represents an application of a Shamir 3-pass specialized to this context.

3. Related Algebraic Constructions

The Algebraic Eraser TM, E, (M )q S, N, II, E, A, B) and its associate key agreement protocol lend themselves naturally to various traditional cate­gorical constructions. Furthermore when we focus on the case of M being a group and S being a (sub)group of automorphisms of the group, a gen­eralized commutator emerges from the E-commuting condition.

As a first example of a categorical construction, we can define the direct product of two algebraic erasers, E1 and E 2 , in a natural way: Next

(M1 )q Sl,N1,II1,E1,Al,B1) X (M2 )q S2,N2,II2,E2,A2,B2) =

((M1 x M 2) )q (Sl x S2), N1 x N 2, III X II2 , E1 X E 2 , A1 X A 2, B1 X B 2 ).

Next, given a submonoid H ::; M which is S invariant, there is a nat­ural sub structure of (M )q S, N, II, E, A, B) to consider; by restricting the functions II and E, and taking suitable intersections we obtain,

(H)q S, N,II lH, E l(NxS)x(H~s),A n H, B n H),

where II lH denotes the restriction of II to H. Finally the concept of a image of an algebraic eraser ™ can be ap­

proached by starting with a homomorphism W : N ----> No and considering the algebraic eraser T M

(M)q S, No, W 0 II, Eo, A, B),

5

where W 0 II, denotes the composite of wand II, and Eo denotes the new action. We leave it to the reader to proceed with this categorical exploration.

When we again restrict ourselves to the case of a group, G and we assume the group S is actually a group of automorphisms of G, S:S Aut(G), then the hypothesis of E-commuting takes the following form. Elements in the subgroups A, B :S G ><I S can be written as

(a,a), (b,(3)

where a, bEG and a, (3 E Aut( G). The function II can be assumed to take the form G ----t G/K, and thus E-multiplication takes the form,

(gK, a) * (h, (3) = (ga(h)K, (3 0 a).

Elements (a, a), (b, (3) E-commute provided identity (1) holds, which in this case takes the form

(aa(b)K,(3 0 a) = ((b(3(a))K,a 0 (3).

The first component of this identity leads naturally to the following gen­eralization of the classical commutator. Given elements x, y E G, and a, (3 E Aut( G), we define the doubly twisted commutator by

C(a,(3,x,y) = x y(3(x-1 )a(y-l).

Clearly when a, (3 = id we are reduced to the familiar classical definition. Letting

n(a,(3,x,y) = a(x) y (3(X)-l

be the associated doubly twisted conjugate, then we have the following analogues of the various classical commutator identities (see [MKS]).

Theorem 3.1. With the notation as above, the following identities hold:

(i) C(a,(3,x,y)-l = C((3-1,a-I,a(y),(3(x))

(ii) C(a, (3, xy, z) = n(id,(3,x, C(a, (3, y, z)) C(id, id,(3(x), a(z))

(iii) C(a,(3,x,yz) = C(id,id,x,y) n(id,a,y,C(a,(3,x,z))

(iv) (identity of H all-Witt type, see [MKSJ)

y-1C (id, a, C( a, a, y, a(x-1)), a(z-l))y

·z-lC (id, a, C( a, a, z, a(y-l)), a 2 (X-I)) z

.a(x-1)C(id, a, C(a, a 2 , a(x), z-l),a(y-l))a(x) = 1

6

Proof. We give selective verifications leaving the remainder to the inter­ested reader. The identity (i) uses the fact that, in general, (a(x))-1 a(x-l):

C( f3 )-1 - (-I)-If3( -1)-1 -1 -1 a, ,x, y - a y x y x

= C(f3-1, a-I, a(y), f3(x))

To prove (ii) we begin with the expression on the right:

n(id, f3, x, C( a, f3, y, z)) C(id, id, f3(x) , a(z))

= xC( a, f3, y, z )f3(x) -1 f3(x )a(x )f3(x )-la(x )-1

= xy zf3(y) -la(z)-1 f3(x) -1 f3(x )a(z )f3(x) -la(z )-1

= C(a,f3,xy,z)

Our third identity is, of course, similar. The final identity is equally ele­mentary: expanding

we obtain

y-1C(id, a, C(a, a, y, a(x-l )), a(z-I))y

oz- I C(id, a, C(a, a, z, a(y-l)), a2(x- 1))z

oa(x-l )C(id, a, C(a, a2, a(x), Z-l), a(y-l) )a(x)

Expanding further we obtain:

7

y-lya(x-1 )a(y -1 )a2(x )a(z-1 )a3 (X-I)

.a2(y )a2(x )a(y-l )a(z )yz-l zy-la (Z-1 )a(y )a2 (X- 1 )a2 (y-l)

·a2 (Z )a(y )a(z-1 )a2(x )za(X-l )a(x )z- la2 (X- 1 )a(z )a(y-l )a2(z-l)

·a3 (X )a(z )a2 (X- 1 )a(y)a(x).

The second and the third lines reduce to the identity, leading to a final cascade of cancellations and the desired identity. 0

To conclude this discussion we again focus on the case of a group G, assuming this time that S ----+ Aut(G) and N coincides with G, i.e.,

II: G ----+ G.

In this case the Algebraic Eraser™ yields a right action of the semi direct product G )<I S on itself (viewed as a set)

(g,S)*(gl,SI) = (gII(Sgl),SSl)

which can be viewed as a twist of the classical right regular representation. Here an element (g, s) is fixed by the element (gl, sd provided that

(g, s) * (gl, sd = (g, s) {=} II(Sgl) = 1, and SI = l.

Thus the stabilizer of the element (g, s) is directly related to the kernel of the endomorphism TI,

Stab(g, s) = {(gl, 1 W gl E ker(TI)}

= {(gl, 1)[ gl E S-"ker(II)}.

In fact the set of stabilizers coincides with the orbit of the action of S (on the left) on ker(II). As with any action the set (see [R]) G)<I S is a disjoint union of the orbits of the action and there is a natural bijection between the orbit of a element (g, s),

Orb(g, s) = {(gTI(Sgl), ssd[(gl, SI) E G )<I S}

and the right cosets

G)<I S mod(Stab(g, s)).

In the case ker(II) is trivial then the stabilizers are themselves trivial. When ker(II) is not trivial, the case of a nonhopfian group comes to mind, the situation becomes for more complex and merits further study. It would be of interest to use a non-hopfian group as a basis for the key agreement protocol introduced in this paper.

8

4. References

[AAGL] Anshel, 1., Anshel, M., Goldfeld, D., Lemieux, S., Key agreement, the algebraic eraser™, and lightweight cryptography, Contemporary Mathematics, 418, 2006.

[AG] Anshel, M., Gaglione, A. M., The search for origins of the com­mutator calculus, Contemporary Mathematics, 421, 2006.

[MKS] Magnus, Wilhelm, Karrass, Abraham, Soli tar , Donald Combinato­rial group theory, Presentations of groups in terms of generators and rela­tions, Reprint of the 1976 second edition. Dover Publications, Inc., Mineola, NY,2004.

[R] Robinson, Derek J. S., A Course in the Theory of Groups, Second Edi­tion, Springer-Verlag, 1995.

Virtually free-by-cyclic one-relator groups. I. *

Gilbert Baumslag t

Department of Computer Science, City College of New York, Convent Avenue at 138th Street, New York, NY 10031 USA

Douglas Troeger+

Department of Computer Science, City College of New York, Convent Avenue at 138th Street, New York, NY 10031 USA

In honor of Anthony Gaglione's 60th birthday

1. Introduction

Many years ago the first author conjectured (Baumslag [2]) that one-relator groups with torsion are residually finite, i.e, the intersection of the sub­groups of finite index is trivial. This paper and its companion (Baumslag, Miller, and Troeger [5]) will be devoted to the exploration of a a much stonger, related conjecture, namely that everyone-relator group with tor­sion is virtually free-by-cyclic, i.e., contains a subgroup of finite index which is an infinite cyclic extension of a free group. Since a finitely gener­ated virtually free-by-cyclic group is residually finite (Baumslag [3]), the first conjecture is a consequence of the second. Moreover finitely generated free-by-cyclic groups are also coherent (Feign and Handel [7]), i.e, their finitely generated subgroups are finitely related. This then impacts on an­other conjecture of the first author, namely that one-relator groups are coherent (Baumslag [4]).

A great deal of effort has gone into proving these conjectures. In partic­ular we draw attention here to the work of Egorov [6], McCammond and Wise [12] and Wise [13] and the references there cited.

• Mathematics Subject Classification. Primary 20F18. t Supported in part by NSF Grants #0202382 and #0430722. tSupported in part by NSF Grant #0430722.

9

10

Now a one-relator group with torsion contains a subgroup of finite index which is torsion-free (Karrass, Magnus and Solitar [9]). Our hope is that one of these torsion-free subgroups is an infinite cyclic extension of a free

group. The Reidemeister-Schreier process [11] allows one to write a presentation

for a subgroup H of a group G given a presentation for G and a collection of coset representatives. If G is finitely presented and H is of finite index then the presentation for H is also finite. If, in addition, K is a normal subgroup of H with H / K infinite cyclic, it is easy enough to present K. However the presentation for K in general will have infinitely many generators and infinitely many relations. The question as to whether such a presentation defines a free group is not easy to answer.

The usual computer implementations of Reidemeister-Schreier cannot be readily adapted to deal with problems of this kind. In view of this, we have implemented a version of Reidemeister-Schreier designed to shed some light on these infinitely generated and infinitely related groups K. Some of the work in this and the refcited paper [5] has been motivated by experimentation using our program.

This experimentation led us to what might appear to be a rather differ­ent effort, namely an expansion of some aspects of computer programming which involves parametric words. Such words arise in rewriting words in an ambient free group as words in the generators of a subgroup using the Reidemeister-Schreier procedure. These words are readily obtained when one carries out the Reidemeister-Schreier procedure by hand in a number of relatively simple examples, but not at the present time by any avail­able computer program. One looks for patterns and then guesses the gen­eral expression. The main objective in this paper is to give an explicit illustration of how this comes about in the case of the one-relator groups Gn =< a, b; (a-1b-1ab)n = 1 >.

In addition, the proof given in Section 4, that a one-relator group where the defining relator has the property that each generator occurs twice with opposite signs is the fundamental group of an orientable surface, is struc­tured as a simple induction which is reflected directly in its computer im­plementation. The result is not new, but the algorithm seems to be.

2. The groups G n

As pointed out to us by Ben Fine, the groups Gn are Fuchsian groups. We recall that subject to a minor restriction, the Fuchsian groups all can be

11

presented in the form

C~" ... C? = 1 > .

Now the finitely generated subgroups of Fuchsian groups are again Fuchsian. Moreover the torsion-free subgroups of finite index are fundamental groups of two-dimensional orient able surfaces and hence, as is well-known, free-by­cyclic. Since, again as is well-known, every Fuchsian group has a torsion­free subgroup of finite index, every Fuchsian group is virtually free-by­cyclic. These facts can be derived by taking advantage of the geometry that underlies the very nature of Fuchsian groups. Direct algebraic proofs of them have also been obtained by Hoare, Karrass and Solitar [8]. It follows therefore that the Gn are virtually free-by-cyclic.

We will give an alternative direct computational proof of this result using the Reidemeister-Schreier process. To do this, we will choose a ho­momorphism of Gn onto a dihedral group and compute an explicit presen­tation of the kernel of this homomorphism. This presentation takes on a parametric form and by using a package of Tietze transformations that was developed for this and other purposes, we shall transform this presentation into a presentation of a fundamental group of an orient able surface. Various aspects of the proof turn out to be of value in our later work, which we will discuss in due turn.

3. The method of Reidemeister and Schreier

Let X be a finite set, and let Fx be the free group with basis X. Let G be a group given by a finite presentation

G = (X; R)

Let H be a subgroup of G, and let T be a right transversal of H in G closed under initial segments. So by definition

• from each coset H 9 of H in G, there is exactly one element, denoted by g, in T;

• if the element w = Yl ... Ye belongs to T, where w is a reduced product of elements of X and their inverses, then Yl ... Yk E T for every k :::; c.

12

We shall refer to 9 as the Schreier representative of the coset H g.

For each element t E T and each generator x EX, we define

a(t,x) = txtx- 1.

Reidemeister and Schreier showed that

S = {a(t,x) It E T and x E X}

is a generating set for H, and further that a presentation for H can be given in terms of these generators. To do so, one defines a term-by-term rewriting function, p, as follows.

For W = y~ly? ... y~P with Yi E X and Ei E {-I, I}, let Wi denote the ith initial segment of w: Wo = 1, and for i > 0, Wi = y~ly~2 ... yfi. Then p(w) is the word obtained from W by replacing each term yfi as follows:

Now H can be given as

where St consists of those a-terms for which tx tx -1 is the identity in Fx , and R = {p(trC 1) I t E T and r E R}.

Henceforth confusing S with S\St, an obvious Tietze transformation yields

H = (S; R)

We will find it useful to recall some elementary facts.

Lemma 3.1. For all wE Fx and all x E Xu X-, wx = W x.

Lemma 3.2. Let T be a Schreier transversal for a subgroup H of G (X; R). For sET and x E X, a(s,x) = 1 if and only if sx E T.

Lemma 3.3. Let T be a Schreier transversal for a subgroup H of G =

(X; R). Suppose s,s' E T and x,x' EX. If neither a(t,x) nor a(t', x') is the identity, then a(t,x) = a(t',x') only ift = t' and x = x'.

13

4. The surface property

A freely reduced word r is a surface word if each generator occurring in r occurs exactly two times - once with exponent -1 and once with exponent 1. Surface words will be said to have the surface property when this makes for better writing.

Two terms with the same generator in a surface word are said to be mates.

Lemma 4.1. Let r = tot1 ... tm be a surface word. Then there are indices

h < i < j < k such that th and tj are mates, and ti and tk are mates.

Proof Choose h < j so that th and tj are mates, and so that j - h is minimal among all separations between mates in r. As r is freely reduced, j > h + 1; further, it is not the case that h = 0 and j = m, by minimality. Consider i between hand j. Where is the mate tk of ti? From the definitions, either k > j or k < h, as otherwise j - h is not minimal. In either case we have a quadruple of the desired kind: (h,i,j,k) with th and tj mates, and ti and tk mates, or (k, h, i, j) with tk and ti mates, and th and tj mates. 0

We exhibit next a series of transformations, suggested by Chuck Miller, transforming a surface word r to a product of commutators.

By the Lemma, we may suppose without loss of generality that r is

woa-1w1b-1w2aw3bw4

Put A-1 = woa- 1, so a = Awo, and substitute into r:

A -lW1b-1w2AwoW3bw4

Next set B = WOW3b, so b- 1 = B-1woW3:

A -lW1 B-1WOW3W2ABw4

Put A = WOW3W2A, so A-1 = A-1wOW3W2. Now r is

A-1WOW3W2W1B-1 ABw4

Finally, let B-1 = WOW3w2w1B-1, so that B = Bwow3w2w1' giving for r

A-1 B- 1 ABwow3w2w1w4

Can we now do the same thing to WOW3W2W1W4? Certainly WOW3W2W1W4

has the surface property if it is freely reduced. If it is not freely reduced, then as free reduction removes mated pairs, the resulting freely reduced word has the surface property. It follows that r may be transformed to a product of commutators, by induction on word length.

14

5. A parametric presentation for the Kernel of ¢: (a,b; [a,b]n) -+ (a,b; a 2,b2n ,abab)

Let

Dn = (a, b; a2, b2n , abab)

and let F[a, bl denote the free group generated by a and b. If

a: F[a,bl-+ Dn

is the homomorphism defined by sending a to a and b to b, then a induces a homomorphism if; : G -+ Dn- In this section we prove

Theorem 5.1. For n > 1, the kernel H of

if;: (a,b; [a,bt) -+ (a,b; a2,b2n ,abab)

has a one-relator presentation on 4n - 2 generators with relator R, parametrized by n. The relator R is a surface word.

We prove this theorem with a series of lemmas and computations. Our main tools are Tietze transformations and the Reidemeister-Schreier procedure. The definition of R is given by Equation 7 in Section 5.3.

To start, we observe

Lemma 5.1. {bi I 0 ~ i < 2n} U {abi I 0 ~ i < 2n} is a right transversal of H closed under initial segments.

5.1. The relators p(b-l[a, b]nbl ).

We need to consider 0 ~ £ ~ 2n - 1. In this section we will show that p(b-f[a, blnbf) is conjugate to p([a, bl n) if £ is even, and to p(b-1[a, blnb) if £ is odd. In detail, we will prove the following:

Lemma 5.2. Let 0 ~ £ ~ 2n - 1. If £ is even, p(b-f[a, blnbf) is conjugate to Va ... Vn-l with

Vk = a(ab2n- 2k , a)-1a(ab2n-(2k+l), b)-la(ab2n-(2k+l), a)a(b2k+1, b) (1)

If £ is odd, p(b-f[a, blnbf) is conjugate to Wa . .. Wn-l where

Wk = a( ab2n-(2k-l), a)-la( ab2n- 2k , b )-la(ab2n- 2k , a)a(b2k, b) (2)

For both the Vk and the Wk, 0 ~ k ~ n -1, and all exponents are computed modulo 2n.

15

As will be seen, each word Vk is the rewriting of the (k+1)8t occurrence of [a, b] in [a, b]n, and each word Wk is the rewriting of the (k + 1)8t occurrence of [a,b] in b-1[a,b]nb.

Before starting the proof, we enumerate some Schreier representatives; the computations depend on the the dihedral relations of Dn.

Lemma 5.3. For all m, we have

(1) am = am (mod 2)

(2) bm = bm(mod 2n) (3) bma = ab-m(mod 2n)

We will most often use the third clause in the form: bma = ab2n-(m (mod 2n)).

5.1.1. The case when C is even

We begin by computing p([a, b]n). For Vk corresponding to the (k + 1)st occurrence of [a, b], as above, one checks

Va = O"(a, a)-10"(ab2n- 1, b)-10"(ab2n- 1, a)O"(b, b)

and further by induction that for 1 ::; k ::; n - 1

Vk = 0"(ab2n- 2k , a)-10"(ab2n-(2k+1), b)-10"(ab2n-(2k+l) , a)0"(b2k+l, b) (3)

Equation (3) is valid for 0 ::; k ::; n - 1 if we agree to reduce exponents modulo 2n.

Next we consider C > O. The last O"-term for the rewritten b-R prefix is 0"(b2n- R, b)-I. Using v~ to denote the subword of p(b-R[a, b]nbR) correspond­ing to the (k + l)st occurrence of [a, b], we have

and, so long as C ;::: 2k + 1,

v~ = 0"( abR- 2k , a) -1 0"(abR-(2k+l) , b) -10"( abR-(2k+l) , a)0"(b2n- ce-(2k+l)) , b)

(4) The next term depends on the parity of C. If C = 2p, the exponent of bin the first term of vp is zero, while if C = 2p + 1, it is not. Letting C = 2p, we have

16

and therefore for k > p

The last term of v~_1 is then 0"(b2n-£-I, b), and it is now evident that the

suffix resulting from b£ is the inverse of the prefix resulting from b-£.

Thus for £ = 2p, p(b-£[a, b]nb£) may be cyclically reduced to yield vb",v~_I' For ° ::::; k < p, the first term of v~ is 0"(ab£-2k,a)-1 and, noting that the expression for v~ when k > p reduces to that for vp when k = p, the first term of v~ for p ::::; k ::::; n - 1 is 0"(ab2n- 2(k- p), a)-I. Thus

the exponents of b in these terms, in the order in which they occur, are

£, £ - 2, ... ,2,0, 2n - 2, ... ,£ + 2

On the other hand, the exponents of b in the first terms of the Vi (not primed), also in the order in which they occur, may be read off from Equa­tion 3 as

0, 2n - 2, ... ,£ + 2, £, £ - 2, ... ,2

As 2 ::::; £ ::::; 2n - 2, and as the remaining three terms of Vi and v~ are determined for each i by the first term of each, it follows that the word vb ... V~_I which survives cyclic reduction is a rotation of Vo ... Vn-I. Thus for £ even, the words p(b-£[a, b]nb£) are all conjugate to p([a, b]n).

5.1.2. The case when £ is odd

Next consider £ = 2p + 1. The sequence vb ... v~_1 begins as for £ even, with v~ given by (4) so long as 2k + 1 < £. What happens for k = p? Now

v~ = O"(ab,a)-IO"(a,b)-IO"(a,a)-IO"(l,b)

and for k > p (in fact, for k >= p)

v~ = dab2n-(2(k- p)-I) , a)-ldab2n- 2(k- p), b)-10"(ab2n- 2(k- p) , a)0"(b2(k-p) , b)

as may be confirmed by induction. Thus the last term prior to those origi­nating with the b£ suffix is

0"(b2«n-l)-p), b)

or

17

So the suffix originating from bl is exactly as before, and again p(b-l[a,b]nbl ) reduces cyclically to yield vb ... V~_lV~ .•• V~_l' Working from the expressions we have given, the exponents of b in the first terms of the respective v~ include every odd integer between 1 and 2n - 1, in this order:

€, € - 2, ... ,3,1, 2n - 1, ... , f + 2

On the other hand, p(b-1[a,b]nb), cyclically reduced, is Wo ... Wn-i, where

Wk = a(ab2n-(2k-l) , a)-la(ab2n- 2k , b)-la(ab2n- 2k , a)a(b2k , b)

The exponents of b in the first terms of the Wk are, respectively,

1, 2n - 1, ... ,3

including every odd integer between 1 and 2n - 1. The order is a cyclic permutation of that resulting from p(b-l[a, b]nbl ).

5.2. The relators p(b-la-1 [a, b]nabl )

Next we prove

Lemma 5.4. Let 0 -::; € -::; 2n - 1. If f is even, p(b-la- 1 [a, b]nabl ) is

conjugate to Xo ... Xn-l with

Xk = a(b2n- 2k , a)-la(b2n-(2k+1), b)-la(b2n-(2k+1), a)a(ab2k+1, b) (5)

If f is odd, p(b-la-1 [a, b]nabl ) is conjugate to Yo· .. Yn-l where

Yk = a(b2n-(2k+1) , a)-la(b2n-(2k+2), b)-la(b2n-(2k+2), a)a(ab2k+2, b) (6)

For both the Xk and the Yk, 0 -::; k -::; n - 1, and all exponents are taken

modulo 2n.

The first f + 1 terms of p(b-ia-1 [a, b]nabi ) are

a(b2n- 1 ,b)-l ... a(b2n- l ,b)-la(abi ,a)-1

The next n terms are x~x~ ... X~_l' with x~ resulting from the (k + 1 )st occurrence of [a, b]. For k such that 2n - (f + 2k + 1) >= 0, x~ is

a(b2n-(H2k), a)-la(b2n-(H2k+1) , b)-la(b2n-(H2k+l), a)a(abH2k+1, b)

If f = 0, these are all of the terms arising from [a, b]n. If f > 0 is even, there exists p such that € + 2p = 2n. Noting that p - 1 then satisfies

18

2n - (£ + 2(p - 1) + 1) >= 0, we can compute the last term of X~_l as u(ab2n- 1, b) and hence

x~ = u(l, a)-lu(b2n-l, b)-lu(b2n-l, a)u(ab, b)

Continuing, for p .:::; k .:::; n - 1, x~ is thus readily seen to be

u(b2n- 2(k- p), a)-lu(b2n- 2(k- p)-1, b )-lu(b2n- 2(k- p)-1, a)u( ab2(k-p)+1 , b)

Taking k = n - 1, we obtain

as the last term preceding the suffix originating from the trailing abR. Hence the suffix of p(b-Ra-1[a, b]nabR) is

u(abR, a)u(b2n- R, b) ... u(b2n-l, b)

and again a cyclic reduction of the relator is possible. As in the proof of Lemma 5.2, x~ ... x~_l is a cyclic permutation of the cyclic reduction of p(a-1[a, b]na), readily computed as Xo ... Xn-l, where

Xk = u(b2n- 2k , a)-lu(b2n-2k-l, b)-lu(b2n-2k-l, a)dab2k+1, b)

If £ is odd, let p satisfy £ + 2p + 1 = 2n, so that

x~ = db, a)-lu(l, b)-lu(l, a)u(a, b)

Subsequent x~ (p .:::; k .:::; n - 1) are therefore given

x~ = u(b2n-(2(k-p)-1), a)-lu(b2n- 2(k- p), b)-lu(b2n- 2(k- p), a)u(ab2(k- p), b)

When k = n - 1, the last term of Wk is

u(abR- 1)

and again the suffix is

u(abR,a)u(b2n- R, b) ... u(b2n-l, b)

Reducing cyclically leaves x~ .. ,X~_l' and we may again compute the or­dered list of exponents of b in the leading terms of each x~ to see that, when £ is odd, the words p(b-Ra-1 [a, b]nabR) are all rotations of the cyclic reduction of p(b-1a- 1 [a, b]nab) given by Yo . .. Yn-l, where

Yk = u(b2n-(2k+l), a)-lu(b2n-(2k+2), b)-lu(b2n-(2k+2) , a)u(ab2k+2, b)

This completes the proof of Lemma 5.4.

19

5.3. Parametric form for the kernel of ¢

It follows from Lemmas 5.2 and 5.4 that a complete set of relators for the kernel H of ¢ is R = {rl' r2, r3, r4} where

rl = p([a, b]n)

r2 = p(b-1[a, b]nb)

r3 = p(a-1[a, bta)

r4 = p(b-1a-1[a, b]nab)

In this section we combine these relators via Tietze transformations into a single relator with parameter n.

To do so, we look more closely at the ri, recalling that each is a product of 4-term words.

For each 0::; k ::; n -1, we abbreviate the four terms of the subword Vk

defined in Lemma 5.2 as follows, from left to right:

v = (J(ab2n- 2k a)-l k,D ,

V = (J(ab2n-(2k+1) b)-l k,l , V = (J(ab2n-(2k+l) a) k,2 ,

V = (J(b2k+l b) k,3 ,

The abbreviations Wk,i, Xk,i and Yk,i for 0 ::; k ::; n - 1 and 0 ::; i ::; 3 are defined analogously, using the expansions given in Lemma 5.2 and Lemma 5.4.

Our first task is to identify those terms occurring in the ri which are freely equal to 1. While including these terms to this point has simplified the derivation and statement of Lemmas 5.2 and 5.4, our subsequent analysis is simplified if we elide them.

From Lemma 3.2 we have

Lemma 5.5. (J (abm , a) is trivial for no value of m, (J (bm , a) is trivial if and only if m = 0, (J(abm , b) = 1 if and only if 0 ::; m < 2n-1, (J(bm , b) = 1 if and only if 0 ::; m < 2n -1, and both (J(l,a) and CJ(l,b) are trivial.

It follows readily that an equivalent set R' = {r~, r~, r~, r~} of relators for H in which no trivial terms appear is given

20

r~ = VO,OVO,l VO,2 Vl,OVl,2 ... Vn -2,OVn -2,2 Vn-l,OVn-l,2Vn-l,3

r~ = WO,OWO,2 ... W n -l,OWn -l,2

r~ = XO,lXO,2 Xl,OXl,2 ..• X n -2,OXn -2,2 Xn-l,OXn-l,2Xn-l,3

I r 4 = YO,OYO,2 ... Yn-2,OYn-2,2 Yn-l,O

We observe next that for 0 ::; i < n - 1, Vi,O = w~i and Vi,2 = wi~\o' and further that Vn-l,O = W;;:-~l 2 and Vn -l,2 = wo6' Similarly, for 0 < i ::; n-1, Xi,O = Yi-l,2 and Xi,2 = Yi,~. Further, XO,2 = 'yo,6. In addition, we may use Lemma 3.3 to demonstrate that no generator occurs more than once in any of the r~, and hence that each (viewed as the equation r~ = 1) may be solved for anyone of its terms.

Accordingly, we may replace each term of ri except VO,l and Vn -l,3 by the inverse of the corresponding term of r~, solve the result for Wo,o, and then substitute this expression for wo,o in r~ to give

R ( -1 (-1 -1) ( -1 -1) 1 = Vn - l ,3 WO,2VO,lWl,O W l ,2W 2,O ( -1 -1) -1 )

.,. W n - 2 2W n-l a W n - 12 , , ,

(WO,2 ... W n -l,OWn -l,2)

in place of ri and r~. Observing that there is no interaction between WO,2

and either of Vn -l,3 or VO,l, we see that Rl is freely reduced as written.

Similarly, r~ and r~ may be replaced by R 2 , obtained via simple Tietze transformations eliminating Yo,o after replacing terms of r~ by inverses of the corresponding terms of r~. In detail,

R (( -1 -1) (-1 -1 ) (-1 -1 ) 2 = YO,2Yl,O . . . Y n -3,2Yn -2,O Y n -2,2Yn -l,O

-1) X n -l,3XO,1

(YO,2 .,. Yn-2,OYn-2,2 Yn-l,O)

again freely reduced as written. (In both equations, parentheses have been inserted for readability.)

Finally, we observe (i) that the unmatched term XO,l in R2 is in fact the inverse of V n -l,3, unmatched in R l , (ii) similarly that X n -l,3 is the inverse of va,!' (iii) that the sets of generators occurring in Rl and R2 are otherwise disjoint, and (iv) that - except for the generators of the unmatched terms -every generator which occurs in Rl occurs precisely twice in R l , once with exponent -1 and once with exponent 1, and similarly for R2. So we may solve Rl for Vn -l,3 and substitute the resulting expression for xO,t in R2 to

21

obtain R with the property that each generator occurs either not at all, or once with exponent -1 and once with exponent 1:

R -1 P -1 -1 -1 Q Q = YO,2 1 Yn -1,O VO,l WO,2 VO,l 1 WO,2 2 YO,2 P2 Yn-1,O (7)

where

P 1 = II ( -1 -1) Yk,OYk,2

1::;k::;n-2

P 2 = II (Yk,OYk,2)

1::;k::;n-2

Q1= II ( -1 -1) W k ,OWk ,2

1::;k::;n-1

Q2 = II (Wk,OWk,2)

1::;k::;n-1

This completes the proof of Theorem 5.1.

6. The kernel K of 1j; : H ---t (z ; ) is free

Consider a one-relator group H = (X; r) where r is a surface word. Let Coo = (z ; ) be the infinite cyclic group. Choose some x E X which occurs in r. If'ljJ : H ---+ Coo is any map sending x to z and all other generators to 1, then, because r has the surface property, 'ljJ extends to a homomorphism. Further,

Theorem 6.1. The kernel K of this homomorphism 'l/J : H ---+ Coo is free.

This theorem coupled with Theorem 5.1 proves that the groups On are virtually free by cyclic.

To prove Theorem 6.1 we apply the Reidemeister-Schreier process to compute a presentation for the kernel K of 'l/J. Observe that a Schreier transversal for K is {x k IkE Z}, and that the Schreier representative for a word W is xk, where k is the exponent sum of x in w.

Assuming first that r has the form

22

where bi, Ci and (i are all either 1 or -1, where by the surface property none of Ul, ... , Urn, VI,"" VP' or WI,"" Wq are x, and where x precedes

-1 . (-k k) Abb . t' 81 8m b c1 8p b x ,we examme p x rx. reVla mg ul ... Urn Y U, VI ... Vp Y V,

d (1 (q b t an WI ... Wq Y W, one compu es

• a(x- j , x) = 1 for all 1 :::; j :::; k

( -k 81 8i - 1 -1 .)-1

• a x U 1 . .. Ui -l Ui ,U, and

( -k 81 8i - 1 .) _ -k . k a x U 1 . .. Ui -l ,U, - X U,x

• a(~,x) = 1 ( -k C1 1 .)-1

• a x UXV1 .. 'Vi ,V, ( -k+l k-l)-1 X ViX and a(x-kuxv~1 .. , Vi-I, Vi-I) = X-k+lViXk-1

• a(x-kuxvx- 1, X)-1 = 1

• a(x-kuxvx- 1wi1 ... Wi 1 , Wi)-1 = (X-kWiXk)-1

• ( -k -IW(1 W(i-1 W) a x uxvx l' .. i-I' i and

a(x- kuxvx- 1wxj , x) = 1 for all 1 :::; j :::; k

Thus each term in the subgroup relator expands to a conjugate. Abbrevi­ating X-kYiXk as Yi,k, we have

We may assume that x was chosen so that, in r, none of V~1, ... , v~n has its mate in v. Noting that V is not empty (r is freely reduced), it follows that there is some letter Vj of V whose mate lies in one of the segments U or w. From the detailed computation one sees then that for all k, p(x-krxk )

includes single occurrences of terms Vj,k and Vj,k+l' That is, the kth relator, for k an arbitrary integer, has one of the forms

or

where ak, bk and Ck are words containing neither Vj,k nor Vj,k+l' Here bk,ck E {I, -I}.

As a similar statement holds if x-I precedes x in r, it follows readily that K is free.

23

8. Virtually locally free by cyclic groups

Having now given our explicit proof that the Gn are virtually free-by-cyclic, we want to present an example which may have some bearing on the con­jecture that one-relator groups with torsion are virtually free-by-cyclic.

Example 6.1. The one-relator group

G =< a, b; (b- Ia2ba-3 )2 = 1 >

has a subgroup K of index 2 with the following properties:

• K contains a normal subgroup L such that K / L is infinite cyclic; • L is locally free; • L is not free.

It seems possible that the group G in this example is not not virtually free by cyclic. Its structure suggests that it may not be. In order to see that G has the properties claimed above, consider the homomorphism ¢ of G onto the group of order 2 generated by t which sends b to 1 and a to t. Then, using the method of Reidemeister and Schreier, it follows that the kernel K of ¢, is of index 2 in G, is generated by x = a2 , y = band z = a-Iba and defined in terms of these generators by the single relation

yxyx- Iz- IxzX-2 = 1.

Now put

Then, it follows from the method of W. Magnus [10] used to solve the word problem for one-relator groups or directly from the method of Reidemeister and Schreier, that the normal closure L of x and z in K is defined by the relations

Now let Lo be the subgroup of L generated by all of the Zi (i E Z) together with Xo. Notice that Xl E Lo and hence X2 E Lo and similarly Xi E Lo for every i ?: O. Notice, by the same token, that if we define L-I to be the subgroup of L generated by all of the Zj and X-I, then Lo ::; L-I with Xo = x=-lz=ix=iz=ix-I' More generally if we now define L-i to be the subgroup of L generated by X-i and all of the Zj for j = 0,1, ... , we find that

24

and 00

We claim that each of the L-i is free on the generators X-i together with all of the Zj. Indeed we can describe this union by generators and relations by taking La to be free on the Zj together with Xa. Then L-I can be taken to be free on the Zj together with X-I and La is identified with a subgroup of L-I by identifying the Zj and identifying Xa with x:: I z=ix=i Z=iX-I. Since x:: I z=tx=t Z=tx-I together with all of the Zj freely generate a free subgroup of the free group L-I' this identification does define an embedding of La in L_ I . The resulting system of generators and relations therefore defines an ascending union of free groups which is nothing more than the description of L in terms of the generators and relations that we obtained before.

Consequently L is indeed a union of free groups and is therefore locally free. It remains to prove that it is not free. In order to see that this is the case, we abelianize L. This abelianization can be described as an abelian group with generators

Xa, X-I, X-2, ...

and the Zj subject to the relations

Thus L abelianized is the direct product of a free abelian group on the Zj

and a copy of the dyadic fractions, i.e., a multiplicative copy of the additive group of rational numbers of the form C 12m . But the abelianization of a free group is free abelian. Hence L is not free. Notice now that KI L is infinite cyclic, so we have proved that G is virtually locally free by cyclic. Since a subgroup of a virtually free by cyclic group is again virtually free by cyclic, G is not virtually free by cyclic.

7. Computational support

As indicated in the Introduction, a computer implementation of Reidemeister-Schreier will be most useful for the application we have in mind here if it can work with the infinitely generated and infinitely related groups which arise routinely when one seeks suitable normal subgroups K of H. Similarly, the usual Tietze transformation package provided for clean­ing up the presentations resulting from Reidemeister-Schreier needs to be extended to provide assistance dealing with possibly infinite relator sets. A

25

third requirement is for machine assistance in searching for patterns of the kind exposed in Section 6.

Our program, exploiting streams and amb-based backtracking, has these capabilities. It is written in Scheme (see the pIt-scheme pages http://www . plt-scheme.org, as well as Abelson and Sussman!), and may be found on our website (http://www.caissny . org).

References

1. Harold Abelson and Gerald J. Sussman. Structure and Interpretation of Com­puter Programs. MIT Press, Cambridge, MA, USA, 1996.

2. Gilbert Baumslag. Residually finite one-relator groups. Bul. Amer. Math. Soc, 73:618-620, 1967.

3. Gilbert Baumslag. Finitely generated cyclic extensions of free groups are resid­ually finite. Bull. Austral. Math. Soc., 5:87-94, 1971.

4. Gilbert Baumslag. Some problems on one-relator groups. pages 75-81. Lecture Notes in Math., Vol. 372, 1974.

5. Gilbert Baumslag, Charles F. Miller III, and Douglas Troeger. Virtually free­by-cyclic one-relator groups. II. Article in preparation.

6. V. Egorov. The residual finiteness of certain one-relator groups. pages 100-121, 1981.

7. M. Feighn and M. Handel. Mapping tori of free group automorphisms are coherent. Preprint, 1997.

8. A. Howard M. Hoare, Abraham K arr ass , and Donald Solitar. Subgroups of finite index of fuchs ian groups. Mathematische Zeitschrift, 120:289 - 298, 1971.

9. A. Karrass, W. Magnus, and D. Solitar. Elements of finite order in groups with a single defining relation. Comm. Pure Appl. Math., 13:57-66, 1960.

10. W. Magnus. Ueber diskontinuierliche gruppen mit einer definierden relation (der freheietssatz). J. Reine Angew. Math., 163:141 - 165, 1930.

11. Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory: Presentations of groups in terms of generators and relations. Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966.

12. J. P. McCammond and D. T. Wise. Coherence, local quasiconvexity, and the perimeter of 2-complexes. Geom. Funct. Anal., 15(4):859-927,2005.

13. Daniel T. Wise. The residual finiteness of positive one-relator groups. Com­ment. Math. Helv., 76(2):314-338, 2001.

SOME CRYPTOPRIMITIVES IN NONCOMMUTATIVE ALGEBRAIC CRYTOGRAPHY

Gilbert Baumslag

Department of Computer Science, City College of New York, New York, N. Y. 10031

Yegor Bryukhov

Department of Computer Science, City College of New York, New York, N. Y. 10031

Benjamin Fine

Department of Mathematics,Fairfield University Fairfield, Connecticut 06430, United States

Gerhard Rosenberger

Fachbereich Mathematik, Universitat Dortmund, 44227 Dortmund, Federal Republic of Germany

Dedicated to A. M. Gaglione on the occasion of his 60th birthday.

Abstract: Recently there has been an active line of research on noncommuta­tive algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-Fine­Xu Modular group scheme use non abelian groups as the basic algebraic object. Some of these encryption methods have been successful and some have been broken. It has been suggested that at this point further pure group theoretic research, with an eye towards cryptographic applications, is necessary. In the present study, the start of large project, we discuss various methods for us­ing noncommutative algebraic objects as methods to develop cryptographic schemes and cryptoprimitives. The project has several parts; the development of general algebraic schemes for cryptoprimitives, the implementation of ap­propriate platforms for these schemes and finally an analysis of the potential security of the schemes.

26

27

1. Introduction

Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The strength of computing machinery has made these techniques theoretically susceptible to attack and hence recently there has been an active line of research to develop cryptosystems and key exchange protocols using noncommutative cryptographic platforms. This line of in­vestigation has been given the broad title of noncommutative algebraic cryptography.

Up to this point the main sources for noncommutative cryptographic platforms have been nonabelian groups. In cryptosystems based on these objects, algebraic properties of the platforms are used prominently in both devising cryptosystems and in cryptanalysis. In particular the nonsolvabil­ity of certain algorithmic problems in finitely presented groups, such as the conjugator search problem, has been crucial in encryption and decryption.

The important sources of nonabelian groups that can be used in cryp­tosystems are combinatorial group theory and linear group theory. Braid group cryptography, where encryption is done within the classical braid groups, is one prominent example. The one way functions in braid group systems are based on the difficulty of solving group theoretic decision prob­lems such as the conjugacy problem and the conjugator search problem (see [AAG] and [KoL]. Although braid group cryptography had initial spectac­ular success, various potential attacks have been identified. Borovik, Myas­nikov, Shpilrain [BMS] and others have studied the statistical aspects of these attacks and have identitifed what are termed black holes in the plat­form groups. These are subsets of the platform groups having the property that choosing cryptographic keys from the group outside of these subsets presents cryptographic problems. In [BFX 1] and [X] potential cryptosys­terns using a combination of combinatorial group theory and linear groups were suggested and a general schema for these types of cryptosystems was given. In [BFX 2] a public key version of this schema using the classical modular group as a platform was presented. A cryptosystem using the the extended modular group SL2 ('1l.) was developed by Yamamura ([Y]) but was subsequently shown to have loopholes ([BG],[S],[HGS]). In [BFX 2] attacks based on these loopholes were closed.

It has been suggested that at this point further pure group theoretic research, and algebraic research in general, with an eye towards crypto­graphic applications, is necessary. In particular, although the present braid

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group cryptosystems may be attackable the basic group theoretic ideas are important. What is then necessary is to look at other (nonabelian) group theoretical methods as well as additional potential platform groups. Along these lines in [BCFRX] an approach was followed based OP a nonabelian group having either a large abelian subgroup or two large subgroups which elementwise commute. Using this idea a general key transport protocol modeled on the classical Diffie-Hellman technique but using a nonabelian group was developed. Several potential groups that could be used as plat­forms were described there, in particular the automorphism group of a free group.

The purpose of this paper is to introduce noncomm'ltative algebraic cryptography to a wider audience and to discuss three general schemes for developing cryptoprimitves in noncommutative cryptographic protocols. Here at first our cryptographic goals are modest. The model we use is that of sending an encrypted message and/or encryption key over public airwaves. Our adversary can view the encrypted message and in addition knows the general technique that we are employing. Our hidden secrets are the actual algberaic objects used and certain parameters of the algebraic objects. This is just a first step and in later work we will expand both the model and the capability of the adversary.

In the present paper we consider two parts - the general scheme itself and potential algebraic platforms where the schemes can be implemented. This is an early part of a general project and a detailed study of cryptanalysis for these methods will follow. All three schemes involve a combination of combinatorial group theory, representation theory and free group rewriting methods. The first two schemes have appeared in various papers so we discuss them only briefly and concentrate on the third.

The first method is a group theoretic version of the classical Diffie­Hellman method and is a generalization of the Anshel-Anshel-Goldfeld scheme. The second method uses a random choice of a subgroup in a linear group followed by a noise factor as the hard problem in group theory needed for encryption. It is a free group polyalphabetic cipher. The final scheme uses the difficulty of factoring in a noncommutative ring. In htis scheme we employ a type of Shamir 3-pass see [CJ]. In particular, relative to this scheme, we explore several different methods to use a formal power series ring R < < Xl, ... , Xn > > in noncommuting variables Xl, ... , Xn as a base to develop cryptosystems. Although R can be any ring we have in mind formal power series rings over the rationals Q. Further we utilize a result of Magnus that a finitely generated free group F has a faithful representation

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in a quotient of the formal power series ring in noncommuting variables. In the next section we describe the basic ideas of free group cryptogra­

phy . In section 3 we describe the general group theoretic Diffie-Hellman method. We briefly mention some potential platforms. In section 4 we de­scribe the general method of Baumsalg-Fine and Xu and the potential plat­forms for this method. In section 5 we describe a general method for using the difficulty of factoring in a noncommutative ring to build a cryptosystem. As a potential platform for this method we suggest the formal power series ring mentioned above. In section 6 we look at some necessary mathematical results in this ring that are needed for crytography.

2. The Basics of Free Group Cryptography

The basic idea in using combinatorial group theory for cryptography is that elements of groups can be expressed as words in some alphabet. If there is an easy method to rewrite group elements in terms of these words and further the technique used in this rewriting process can be supplied by a secret key then a cryptosystem can be created. The simplest example is perhaps a free group cryptosystem. This can be described in the following manner. We will use the books by Magnus, Karrass and Solitar [MKS] or Baumslag [B] as standard references for material on combinatorial group theory. All of the proposed schemes in this paper are based in a general sense on free group cryptography.

Consider a free group F on free generators Xl, ""xr' Then each element gin F has a unique expression as a word W(XI, ... , x r ). Let WI, ... , W k with Wi = Wi (Xl, ... , x r ) be a set of words in the generators Xl, ... , Xr of the free group F. At the most basic level, to construct a cryptosystem, suppose that we have a plaintext alphabet A. For example suppose A = {a, b, ... } are the symbols needed to construct meaningful messages in English. To encrypt, use a substitution ciphertext

That is

Then given an word W(a, b, ... ) in the plaintext alphabet form the free group word W(WI , W2, .... ). This represents an element 9 in F. Send out 9 as the secret message.

In order to implement this scheme we need a concrete representation of 9

and then for decryption a way to rewrite 9 back in terms of WI, ... , Wk. This

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concrete representation is the idea behind homomorphic cryptosystems (see the article of Grigoriev and Ponomarenko [GP]).

The decryption algorithm then depends on a very important idea that we will need later and which is known as the Reidemeister-Schreier rewriting process (see [MKSj for full details). Assume WI, .... Wk are free generators for some subgroup H of F. A Schreier transversal for H is a set {hI, ... , hi> ... } of (left) coset representatives for H in F of a special form (again see [MKS] for particular details). Any subgroup of a free group has a Schreier transversal. The Reidemeister-Schreier process allows one to construct a set of generators WI, ... , Wk for H by using a Schreier transver­sal. Further given the Schreier transversal from which the set of generators for H was constructed, the Reidemeister-Schreier Rewriting Process allows us to algorithmically rewrite an element of H. Given such an element expressed as a word W = W (Xl, ... . x r ) in the generators of F this algorithm rewrites W as a word W*(WI' ... , Wk) in the generators of H. The actual algorithm is described in detail in [MKS] and [B].

An important feature of the Reidemeister-Schreier rewriting process is that to rewrite a word

where f.ij E {-1,1} and i j E {l,2, ... r}, the algorithm rewrites letter by letter from left to right (see [MKS] for complete details). This was an important feature of the Baumslag-Fine-Xu Modular group scheme (see [BFX 1,2]).

One of the earliest descriptions of a free group cryptosystem as well as a homomorphic version of it was in a paper by W. Magnus in the early 1970's [M]. Pure free group cryptosystems are subject to various attacks especially length-based attackes and can be broken easily (see [GP] and [St]). Therefore enhancements must be made to free group cryptosystems to provide some security.

3. A General Schema for Nonabelian Group Diffie-Hellman

The groundwork for the present noncommutative algebraic cryptogra­phy was laid in two seminal papers by Anshel-Anshel-Goldfed [AAG] in 1999 and Ko-Lee[KoL] in 2000. The resulting proposed cryptographic schemes can be considered as group theoretic analogs of the number theory based Diffie-Hellman method. Both methods are entirely general. However Anshel-Anshel-Goldfeld proposed using the classical braid groups and cer­tain fixed parameters as a particular platform. Cryptanalysis has been done

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on both methods but only relative to the platforms and parameters in these platforms that the developers suggested.

In [BCFRX] the following generalization of the AAG scheme was de­scribed. Suppose that G is a finitely presented group that can be repre­sented in a nice way - either as a matrix group or as words relative to a nice presentation. Further suppose that G has two large subgroups A I ,A2

that commute elementwise. Alternatively we could use one large abelian subgroup A of G. The meaning of large is of course hazy but relative to the encryption scheme it means that within G it is difficult to determine when an arbitrary element is in Al or A2 (or A) and further Al and A2 (or A) is large enough so that random choices can be made from them. For our purposes large can mean that Al and A2 both contain nonabelian free subgroups.

Now suppose that Bob wants to communicate with Alice via an open air­way. The message (or the secret key telling them which encryption system to use) is encoded within the finitely generated group G with the proper­ties given above. The two subgroups AI, A2 which commute elementwise are kept secret by Bob and Alice. Al is the subgroup for Bob and A2 the sub­group for Alice. Bob wants to send the key W E G to Alice. He chooses two random elements B I , B2 E Al and sends Alice the message (in encrypted form) BI W B2. Alice now chooses two random elements GI , G2 E A2 and sends GIBI WB2G2 back to Bob. These messages appear in the represen­tation of G and hence for example as matrices or as reduced words in the generators so they don't appear as solely concatenation of letters. Since Al commutes elementwise with A2 we have

Further since Bob knows his chosen elements BI and B2 he can multiply by their inverses to obtain GI WG2 which he then sends back to Alice. Since Alice knows her chosen elements GI , G2 she can multiply by their inverses to obtain the key W. It is assumed that for each message Bob and Alice would choose different pairs of random elements from either Al or A 2 . This method is a variation of a suggestion of Shamir now known as a Shamir 3-Pass or Three Pass Protocol (see [CJ]). We will apply a similar technique in a ring theoretic setting in section 5.

In [BCFRX] several potential platform groups are suggested. These in­clude the full automorphism group of a finitely generated free group, the matrix group SL( 4, Z) and the surface braid groups. Shpilrain and Ushakov [SU] used this method employing Thompson's group F as a platform. A

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length based attack on their system was attempted by Tsoban [T]. Further work on this method in the surface braid groups is being done by Camps[C].

4. Polyalphabetic Free Group Cryptosystems

In [BFX 1] the following general encryption scheme using free group cryp­tography was described. A further enhancement was discussed in [BFX 2].

We start with a finitely presented group

G=<XIR>

where X = {Xl, ... , xn} and a faithful representation

p: G -t G.

G can be anyone of several different kinds of objects - linear group, per­mutation group, power series ring etc.

We assume that there is an algorithm to re-express an element of p( G) in G in terms of the generators of G. That is if 9 = W(Xl' ... ,xn ... ) E G where W is a word in these generators and we are given p(g) E G we can algorithmically find 9 and its expression as the word W(Xl' .. xn).

Once we have G we assume that we have two free subgroups K, H with

He KeG.

We assume that we have fixed Schreier transversals for K in G and for H in K both of which are held in secret by the communicating parties Bob and Alice (see [B] for a description of Reidemeister-Schreier). Now based on the fixed Schreier transversals we have sets of Schreier generators constructed from the Reidemeister-Schreier process for K and for H.

kl' ... km ,... for K

and

hl, ... , ht ,... for H.

Notice that the generators for K will be given as words in Xl, ... , Xn the generators of G while the generators for H will be given as words in the generators kl' k2' ... , for K. We note further that Hand K may coincide and that Hand K need not in general be free but only have a unique set of normal forms so that the representation of an element in terms of the given Schreier generators is unique.

We will encode within H, or more precisely within p( H). We assume that the number of generators for H is larger than the set of characters within

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our plaintext alphabet. Let A = {a, b, c, ... } be our plaintext alphabet. At the simplest level we choose a starting point i, within the generators of H, and enclode

a -t hi, b -t hHI' .... etc.

We now again use a Shamir 3-Pass specialized to this ring-theoretic setting. Suppose that Bob wants to communicate the message W(a, b, c ... ) to Alice where W is a word in the plaintext alphabet. Recall that both Bob and Alice know the various Schreier transversals which are kept secret between them. Bob then encodes W(hi' hi+1"') and computes in G the element W(p(hi ), p(hi+1), .. ) which he sends to Alice. This is sent as a matrix if G is a linear group or as a permutation if G is a permutation group and so on.

Alice uses the algorithm for G relative to G to rewrite W(p(hi ), P(hHd, .. ) as a word W*(XI' ... x n ) in the generators of G. She then uses the Schreier transversal for K in G to rewrite using the Reidemeister-Schreier process W* as a word W**(kI' ... , ks .. ) in the generators of K. Since K is free or has unique normal forms this expression for the element of K is unique. Once Alice has the word written in the generators of K she uses the transversal for H in K to rewrite again, using the Reidemeister-Schreier process, in terms of the generators for H. She then has a word W***(hi' hHI' ... ) and using hi -t a, hHI -t b, ... decodes the message.

In [FBX 1,2] an inplementation of this process was presented that used for the base group G the classical modular group M = PSL(2,7l,). Further it was a polyalphabetic cipher which was secure.

The system in the modular group M was presented as follows. A list of finitely generated free subgroups HI, .... , Hm of M is public and presented by their systems of generators (presented as matrices). In a full practical implementation it is assumed that m is large. For each Hi we have a Schreier transversal

hI,i, ... , ht(i),i

and a corresponding ordered set of generators

WI,i, ... , W m(i),i

constructed from the Schreier transversal by the Reidemeister-Schreier pro­cess. It is assumed that each m( i) > > I where I is the size of the plaintext alphabet, that is each subgroup has many more generators than the size of the plaintext alphabet. Although Bob and Alice know these subgroups in

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terms of free group generators what is made public are generating systems given in terms of the polynomials in noncommuting variables.

The subgroups on this list and their corresponding Schreier transversals can be chosen in a variety of ways. For example the commutator subgroup of the Modular group is free of rank 2 and some of the subgroups Hi can be determined from homomorphisms of this subgroup onto a set of finite groups.

Suppose that Bob wants to send a message to Alice. Bob first chooses three integers (m, q, t) where

m = choice of the subgroup Hm

q = starting point among the generators of Hm

for the substitution of the plaintext alphabet

t = size of the message unit .

We clarify the meanings of q and t. Once Bob chooses m, to further clarify the meaning of q, he makes the substitution

Again the assumption is that m( i) > > l so that starting almost anywhere in the sequence of generators of Hm will allow this substitution. The message unit size t is the number of coded letters that Bob will place into each coded integral matrix.

Once Bob has made the choices (m, q, t) he takes his plaintext message W(a, b, ... ) and groups blocks of t letters. He then makes the given substi­tution above to form the corresponding matrices in the Modular group;

We now introduce a random noise factor. After forming T I , ... , Ts Bob then multiplies each Ti on the right by a random polynomial in F say RT; (

different for each Ti). The only restriction on this random polynomial RT;

is that there is no free cancellation in forming the product TiRT;. This can be easily checked and ensures that the freely reduced form for TiRT; is just the concatenation of the expressions for Ti and RT;. Next he sends Alice the integral key (m, q, t) by some public key method (RSA, Anshel-Goldfeld etc.). He then sends the message as s random projective matrices

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Hence what is actually being sent out are not elements of the chosen sub­group Hm but rather elements of random right cosets of Hm .. The purpose of sending coset elements is two-fold. The first is to hinder any geometric attack by masking the subgroup. The second is that it makes the resulting words in the the Modular Group generators longer - effectively hindering a brute force attack.

To decode the message Alice first uses public key decryption to obtain the integral keys (m, q, t). She then knows the subgroup H m , the ciphertext substitution from the generators of Hm and how many letters t each matrix encodes. She next uses the algorithms described in section 2 to express each TiRTi in terms of the free group generators of F say WTi (Yl,", Yn). She has knowledge of the Schreier transversal, which is held secretly by Bob and Alice, so now uses the Reidemeister-Schreier rewriting process to start expressing this freely reduced word in terms of the generators of Hm. Recall that Reidemeister-Schreier rewriting is done letter by letter from left to right. Hence when she reaches t of the free generators she stops. Notice that the string that she is rewriting is longer than what she needs to rewrite in order to decode as a result of the random matrix RTi' This is due to the fact that she is actually rewriting not an element of the subgroup but an element in a right coset. This presents a further difficulty to an attacker. Since these are random right cosets it makes it difficult to pick up statistical patterns in the generators even if more than one message is intercepted. In practice the subgroups should be changed with each message.

The initial key (m, q, t) is changed frequently. Hence as mentioned above this method becomes a type of polyalphabetic cipher. Polyalphabetic ci­phers have historically been very difficult to decode [H].

5. Factoring in Noncommutative Rings

The following method is a further extension to the noncommutative ring setting of the nonabelian group theoretic Diffie-Hellman system. It uses the difficulty of factoring within some noncommutative rings together with the presence of a large and computable unit group.

We suppose that we have a ring R with a large unit group U(R). By large we mean that U(R) contains a nonabelian free subgroup so that random choices can be made from U(R). We suppose further that there is no factoring algorithm in R. We also suppose that U(R) is computable and one knows that r E u(R) then r-1 can be found. Suppose that Bob wants to send a message to Alice. Encoding is done within R so that elements of R represent messages. Bob wants to send the message r E R to Alice. He

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randomly chooses an e E U(R) and sends Alice reo Alice randomly chooses f E U(R) and sends back fre. Bob knows (but presumably an attacker cannot figure out) e- l so he forms free- l = fr and sends this back to Alice. Alice applies f- l to get the message r.

As a platform for this encryption method we propose the use of the ring of formal power series

R« Xl, "',Xn » over a ring R in noncommuting variables Xl, ""xn' Although this can be done in an even more general context, for this study we will concentrate on rational formal power series, that is we consider the ring R to be the field of rational numbers Q.

Throughout the rest of this paper we let

H=Q«XI, ... ,Xn »

be the formal power series ring in noncommuting variables Xl, ... , Xn over Q. One of our primary tools for developing encryption methods will be based upon a faithful representation of a finitely generated free group within a quotient of H. This representation was developed by W.Magnus [MJ and is now known as the Magnus representation. If n 2: 2 this then provides free subgroups of all countable ranks within this quotient of H. Further by imposing additional relations we can obtain representations of free nilpo­tent groups. This method can be applied using the ring H as the platform. The power d defining H is a shared secret. Encryption is done in a poly­nomial in the noncommuting variables. This encryption can be done in a variety of ways. The simplest is perhaps the following. The coefficients of our polynomials are rational numbers. Code letters by rational numbers and the message is read off from the coefficients.

Let d > 1 be an integer and impose the relations

x1 = x~ = ... = x~ = 0

on H. We call the resulting quotient H. Notice that the elements of H are polynomials of degree < d in the non­

commuting variables Xl, ... , X n . The faithful representation of a free group is given in terms of the monomials

al = 1 + Xl, a2 = 1 + X2, ... , an = 1 + X n .

Notice that in the formal power series ring H we have the well known expansion

1 _ 2 3 -1-- - 1 - Xi + Xi - Xi + ......

+Xi

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Therefore each ai is invertible within H and hence invertible in H. Within H however the inverse is a polynomial of degree < d and so within H

1 _ 2 3 d-l d-l -1-- - 1 - Xi + Xi - Xi + ... + ( -1) x· . +~ I

Therefore each ai is in the unit group U(H) of H and therefore the set {aI, ... , an} generates a multiplicative subgroup of U(H). Note also that if d, the defining power, is kept secret, then inverses are unknown. Further we can show that the unit group U(H) consists of those polynomials with non-zero constant term.

Suppose that Bob wants to send Alice the message T E Q[[XI, .. , xnll where xf = 0 for all i. Let R = T + S where S is an arbitrary polynomial with only powers higher than d. Bob chooses a random element of the unit group W. He sends Alice RW. Bob knows the inverse of W. Alice chooses another random V of the unit group and sends Bob back V RW. Bob multiplies by W- l and sends Alice V R from which Alice recovers R. Since she knows d she cancels all powers higher than d to obtain the message T. An attacker would need to factor RW and know the defining power d to attack the message. We mention that this scheme would not work if we restricted W to be in the Magnus free group within H. We will show in the next section that if Bob sends RW and Alice sends back V RW then as V can be peeled off and subsequently W can be peeled off so the attacker can get the message R.

6. Formal Power Series Rings and the Magnus Representation

We now describe some necessary algorithmic material needed to use the rings Hand H as cryptographic platforms. Recall that

H = Q < < Xl, ... , Xn > >

is the formal power series ring in noncommuting variables Xl, ... , Xn over Q and H is the resulting qotuient ring found by letting xf = 0 for all i. This D is kept secret in cryptography. One of our primary tools for developing encryption methods will be based upon the Magnus representation of a finitely generated free group within a quotient of H. If n ~ 2 this represen­tation provides free subgroups of all countable ranks within this quotient of H. Further by imposing additional relations we can obtain representations of free nilpotent groups.

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We first describe the Magnus representation and give a proof. The proof will lead us to two algorithms for describing when certain polynomials lie in the image of this representation. Further we describe the unit group of this quotient.

First let d > 1 be an integer and impose the relations

x~ = x~ = ... = x~ = 0

on H. We call the resulting quotient H. Notice that the elements of H are polynomials of degree < d in the non­

commuting variables Xl, ... , X n . The faithful representation of a free group is given in terms of the monomials

Notice that in the formal power series ring H we have the well known expansion

1 2 3 -- = 1 - Xi + Xi - Xi + ...... 1 +Xi

Therefore each Q:i is invertible within H and hence invertible in H. Within H however the inverse is a polynomial of degree < d and so within H

1 2 3 d-l d-l -- = 1 - Xi + Xi - Xi + ... + ( -1) Xi . 1 +Xi

Therefore each Q:i is in the unit group U(H) of H and therefore the set {Q:I, ... , Q:n} generates a multiplicative subgroup of U (H). Note also that if d, the defining power, is kept secret, then inverses are unknown

Magnus' result is the following.

Theorem 6.1. The elements

freely generate a subgroup of U(H). Therefore the map given by

provides a faithful representation of the free group on YI, ... , Yn into H.

We present a proof, since as mentioned, the proof will lead us to an algorithm necessary for our encryption methods.

Proof. Notice from the comment above that each Q:i is invertible within H. Therefore each Q:i is in the unit group U(H) of H and therefore the set

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{ ai, ... , an} generates a multiplicative subgroup of U (H). We show that no nontrivial freely reduced word in the ai can be the identity and hence the group they generate must be a free group.

From the binomial expansion we have for any non-zero integer n, posi­tive or negative,

(1 + ai)n = 1 + nai + terms in higher powers.

Now let

be a freely reduced word in the ai with each Inil ?': 1 and aij "I=- ai}+1 for j = 1, ... , k - 1. For later reference we call k the block length. In the ring H we then have

and hence

W(al' ... , an)

= (1 + n1xi1 + higher powers in Xi1) ... (1 + nkxik + higher powers in Xik)

The variables are noncommuting, so that in analyzing this product we see that there is a unique monomial term of maximal block length k where each Xij appears to the power 1. That is there is a unique monomial term

We stress here that this is of maximal block length since this will be im­portant in the subsequent algorithm.

Since each ni "I=- 0 this term must appear and therefore W(al' ... , an) "l=­

I. It follows that the group generated by 001, .. , an is freely generated by them. D

The proof of the faithfulness of the Magnus representation leads us to several algorithms for dealing with the image in the power series ring. We will employ these algorithms in our cryptosystems. For the remainder of this section we will let F denote the free subgroup of H generated by the

The first algorithm provides a method, given a polynomial in H, which is written in polynomial form, that we know to be in F, to write its unique free group decomposition. That is given

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a polynomial in the noncommuting variables Xl, .'" Xn that we know to be in F to rewrite f as

In general there is no factoring algorithm in H. For any monomial Xi, ... Xik in H we call k the block length of the

monomial in analogy with that of a free group word.

Theorem 6.2. (Algorithm to Recover the Free Group Decomposition of Elements in F). Suppose f = f(XI, ... , xn) E H and it is known that f E

F. There is an algorithm that rewrites f in terms of the free generators all ... , an, that is the algorithm uniquely expresses f as a free group word

The algorithm works as follows: Step 1: In f locate the monomial nXil ... Xik of maximal block length

where n E Z \ {O}, each variable that appears in f appears in this monomial, and each variable is to the power 1. This k gives the block length for the corresponding free group word. Further the free group word must have the form

with each ni a divisor of n. Step 2: For each divisor ni of n both positive and negative sequentially

form (1 + Xi,) -n, f. In exactly one such product the maximal block length will be k - 1 and there will be a unique monomial of block length k - 1 containing each variable in f except perhaps Xi, and each to the power 1. We then have

where !I is also in F. Step 3: Continue in this manner until we reach the identity. The free

group decomposition of f is then

f = (1 + X· )nl ... (1 + X· )nk = a':' ... a nk 1.1 1.k 1.1 tk .

Proof. Since we know that f E F we know that there is a unique free group decomposition

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Hence, as in the proof that the representation is faithful, there is a unique monomial

of maximal block length where n E Z \ {O}, each variable that appears in f appears in this monomial, and each variable is to the power 1. Again as in the proof of Theorem 6.2, k gives the block length for the corresponding free group word.

Now, since the free group representation is unique we have for each divisor ni of n

(1 + xil)-ni f = (1 + Xil)n1-n i ..• (1 + Xikt k.

Hence only for ni = nl will this term cancel. Hence there is exactly one such divisor such that (1 + Xi)-ni F will now have maximal block length k - 1 and have a unique monomial of the prescribed type. It follows then that one and only one such product will reduce f to a word of shorter block length. 0

A modification of the above algorithm can be used to determine if a general element of H is actually in F.

Theorem 6.3. (Algorithm to Determine if f E H is in F). Suppose

f = f(XI, ... ,xn ) E H.

There is an algorithm that determines whether or not f E F and if it is, rewrites f in terms of the free generators aI, ... , an. The algorithm works as follows:

Step 1: If the constant term of f i- 1 then f rJ. F. FUrther if f has any nonintegral coefficients then f rJ. F.

Step 2: Assume f passes Step 1. If f does not contain a unique monomial nXil ... Xik of maximal block length in f where n E Z \ {O}, each variable that appears in f appears in this monomial, and each variable is to the power 1 then f rJ. F.

Step 3: Suppose f passes Steps 1 and 2. Then in f locate the mono­mial with the characteristics described in Step 2. If f E F then k gives the block length for the corresponding free group word. Further the free group word must have the form

with each ni a divisor of n.

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Step 4: For each divisor ni of n both positive and negative sequentially form (1 +Xil)-n1 f. If in such product the maximal block becomes k -1 and there is a no new monomial having the described characteristics above then f ¢:. F. Otherwise continue.

Step 5: If eventually we arrive at the identity then f E F and the procedure yields the free product decomposition of f.

Proof. The proof follows in exactly the same manner as the proof of The­orem 6.2. 0

For certain cryptographic applications we need the full unit group U(H) of H. Over Q it can be described as those polynomials with nonzero constant term.

Theorem 6.4. The unit group U(H) over Q consists precisely of those polynomials with nonzero constant term.

Proof. There are two ways to look at the proof of this. Algebraically sup­pose the defining power is d > 1 and P(x) E H with nonzero constant term. Then P(x) is relatively prime to the polynomials xt and so is invertible in the factor ring in the standard way.

Analytically if P(x) E H with nonzero constant term let P*(x) be the corresponding polynomial in H. Then P* (x) can be made into part of a convergent power series P**(x) in

C« Xl, ... ,Xn » .

Since P**(O) =I- 0 this power series is analytic at 0 and so its inverse is analytic at 0 and so has a convergent power series around 0 say Q**(x). The image of Q**(x) in H would then be the inverse of P(x).

Conversely if P(x) E H is invertible it must have nonzero constant ten)[J

Before we continue we mention one final item concerning multiplication within H. In general there is no factoring algorithm. However if f E H is known and g = fe with e E F is known then we can find e. We say that e can be peeled off f e. The algorithm to do this is essentially the same as the above two algorithms. We briefly explain. Suppose we are given f and fe. Then in fe there is a unique monomial extending the monomials in f exactly as in the proof of Theorem 6.2. By identifying this monomial we can find the free group decomposition of e and hence find e.

The rings H and its quotient H, together with the Magnus representa­tion, provide a very flexible platform for doing cryptography. First of all the

43

unit group U(R) contains a free group via the Magnus representation. Fur­ther there is no factoring algorithm within H. Keeping the defining power d secret makes determining inverses open only to those who know d. Hence this ring provides an ideal algebraic platform for the ring theoretic Diffie­Hellman method described in the last section. The actual method goes as was described there which we briefly repeat.

Suppose that Bob wants to send Alice the message T E Q[[Xl, '" xnll where x~ = 0 for all i. Let R = T + S where S is an arbitrary polynomial with only powers higher than d. Bob chooses a random element of the unit group W. He sends Alice RW. Bob knows the inverse of W. Alice chooses another random V of the unit group and sends Bob back V RW. Bob multiplies by W-l and sends Alice VR from which Alice recovers R. Since she knows d she cancels all powers higher than d to obtain the message T. An attacker would need to factor RW and know the defining power d to attack the message.

Further the unit group U(R) of H contains free subgroups. Via the algorithms described in this section we can move back and forth between the free group F and the representing polynomials. Hence this can be used as a platform for the free group BFX polyalphabetic cipher.

7. Relations on the Variables

It was shown in [GB 2] that by imposing further relations on the variables, free nilpotent groups of all possible class size can also be embedded in quotients of H. This was used in [GB 2] to prove certain results concerning equations in free groups. This can be used further for encryption purposes. By imposing nilpotency relations on some of the variables in the power series, but not all, and keeping the relations secret a further level of security is imposed. This procedure is under development ([BBFGR]).

8. References

[AAG] LAnshel,M.Anshel,D.Goldfeld, An Algebraic Method for Public Key Cryptography, Math.Res. Lett, 6, 1999, 287-291 Springer Verlag

[GB 1] G.Baumslag, Topics in Combinatorial Group The-ory,Birkhauser 1993

[GB 2] G. Baumslag, Residual Nilpotence and Relations in Free Groups J. Algebra, 2, 1965, 271-285

[BFX 1] G. Baumslag, B.Fine, and X.Xu, Cryptosystems Using Linear

44

Groups Appl. Alg. in Engineering,Communication and Comput­ing, 17, 2006, 205-217

[BFX 2J G. Baumslag, B.Fine, and X.Xu, A Proposed Public Key Cryp­tosystem Using the Modular Group Cont. Math, 421, 2007, 35-44

[BCFRXJ G. Baumslag,T.Camps, B.Fine,G.Rosenberger and X.Xu, Design­ing Key Transport Protocols Using Combinatorial Group Theory, Cont. Math., 418, 2006, 35-43

[CJJ J.Clark and J.Jacob, A Survey of Authentication Protocol Literature; Version 1, No. 1997 , see www.lsv.ens-cachan.fr/spore/shamir.pdf

[FJ B.Fine, The Algebraic Theory of the Bianchi Groups, Marcel Dekker, 1990

[GPJ D.Grigoriev and 1. Ponomarenko, Homomorphic Public-Key Cryp­tosystems Over Groups and Rings Quaderni di Matematica, 2005

[HJ P.Hoffman, Archimedes Revenge Fawcett-Crest, 1988

[HGSJ C.Hall, 1. Goldberg, B. Schneier, Reaction attaacks Against Several Public Key Cryptosystems Proceedings of Information and Commu­nications Security ICICS 99, Springer-Verlag, 1999, 2-12

[KoLJ K.H. Ko,J.Lee,J.H. Cheon,J.W. Han, J.Kang, C.Park, New Public­Key Cryptosystem Using Braid Groups, inAdvan Advances in Cryptol­ogy - CRYPTO 2000 Santa Barbara CA, - Lecture Notes in Computer Science, Springer 1880, 2000 166-183

[KoJ N.Koblitz, Algebraic Methods of Cryptography, Springer, 1998

[MJ W. Magnus, Rational Representations of Fuchsian Groups and Non­Parabolic Subgroups of the Modular Group, Nachrichten der Akad Gottingen, 1973, 179-189

[MKS] W. Magnus, A. Karass and D. Solitar Combinatorial Group The­ory, Wiley Interscience,New York, 1968

[StJ R. Steinwandt, Loopholes in two public key cryptosystems using the modular groups preprint Univ. of Karlsruhle, 2000

[X] Xiaowei Xu, Cryptography and Infinite Group Theory, Ph.D. Thesis, CUNY,2006

[YJ A.Yamamura, Public Key cryptosystems using the modular group, Lecture Notes in Comput. Sci., 1431, 1998, 203-216

On the derived subgroups of the free nilpotent groups of finite rank

Russell D. Blyth

Department of Mathematics and Computer Science Saint Louis University, St. Louis, MO 63103, USA

Email address: blythrd@slu.edu

Primoz Moravec

Fakulteta za matematiko in fiziko Univerza v Ljubljani

Jadmnska 19, 1000 Ljubljana, Slovenia Email address:primoz.momvec@fmf.uni-lj.si

Robert Fitzgerald Morse

Department of Electrical Engineering and Computer Science University of Evansville

Evansville IN 47722 USA Email address: rfmorse@evansville.edu

URL: faculty. evansville. edu/rm43

Dedicated to Tony Gaglione on his Sixtieth Birthday

We provide a detailed structure description of the derived subgroups of the free nilpotent groups of finite rank. This description is then applied to computing the nonabelian tensor squares of the free nilpotent groups of finite rank.

Keywords: Free nilpotent group, Derived subgroup, Nonabelian tensor square 2000 Mathematics Subject Classification: 20F18, 20J99

1. Introduction

A systematic structure description of the subgroups of the free nilpotent groups is given in S. Moran's paper "A subgroup theorem for free nilpotent groups" 1 that is based on the work of Gol' dina. 2 Moran's result is neces­sarily general and does not provide a detailed structure description for any specific subgroup of a free nilpotent group, such as its derived subgroup.

The purpose of this paper is to provide a detailed structure descrip-

45

46

tion of the derived subgroups of the free nilpotent groups of finite rank. The motivation for this investigation is that the derived subgroup of a free nilpotent group of class c + 1 and rank n is isomorphic to the nonabelian exterior square of the free nilpotent group of class c and rank n. Moreover, the results presented in this paper give complete structure descriptions of the nonabelian tensor squares of free nilpotent groups of finite rank using a result from Blyth et al. 3

We fix our notation. Let Fn be the free group of rank n with generators II, ... , fn and denote the free abelian group of rank n by F:;b. Let Cn be a fixed Hall basic sequence of commutators in the free generators of Fn. The weight of the commutator Ci E Cn is denoted by Wi. We denote the subsequence of commutators of Cn whose weight is at most w by Cn,w. The number of commutators in Cn,w \Cn,w-1 is denoted by M(n, w). The subset of simple left normed commutators in Cn of weight at most w is denoted by Sn,w. Let Nn,c = Fnhc+1(Fn) be the free nilpotent group of rank nand class cgenerated by gl, ... , gn. Denote by 'Dn,c the derived subgroup ofNn,c. The elements of Cn,c map to Nn,c via the natural homomorphism Fn ----t

Fnhc+1 (Fn) = Nn,c. With slight abuse of notation we identify elements of Cn,c as the same as their images in Nn,c.

The group Nm,w, which will be used several times in this paper, is defined as follows. We first fix a free nilpotent group K. Given m 2:: 2 and w 2:: 3, let

ifm = 2

if m 2:: 3,

where s = ISm,w-2 \ Sw,ll. Let k1, ... , ks be the free generating set for K. Fix a bijection f3 from the set {1, ... , s} to Sm, w- 2 \ Sm,l. Associate a weight Wk i to each generator ki of K via f3 by setting Wk

i to be the weight

of the simple left normed commutator f3(i). Then for m 2:: 2 and w 2:: 3, define

Nm,w = K/R,

where R = ([k i , kj ] I Wki + Wk j > w). The group Nm,w has a minimal cardinality generating set with s = ISm,w-2 \ Sw,ll generators.

Our main result is the following description of the derived subgroup of a free nilpotent group of finite rank.

Theorem 1.1. Let Nn,c be the free nilpotent group of class c 2:: 1 and rank

n 2:: 1. If n = 1 or c = 1 then Nn,c is abelian and 'Dn,c is trivial. If n > 1

and c = 2 then Vn,c is free abelian of rank M(n,2) c> 2 then

where f = ISn,c \ Sn,c-21·

47

G). If n > 1 and

In Section 2 we prove Theorem 1.1 and provide formulas for ISn,c \ Sn,c-21 and ISn,c \ Sn,ll. These formulas allow us to restate Theorem 1.1, giving an explicit rank of the free abelian factor of Vn,c and for the minimal number of generators for Nn,c (Theorem 2.3).

The nonabelian tensor square G I8i G of a group G was introduced by Brown and Loday4 following the ideas of Dennis5 and Miller6 and is of topological significance. In Section 3 we give a brief exposition of the non­abelian tensor square of a group and use the formulas found in Section 2 to give a full description of Nn,c I8i Nn,c' We apply Theorem 1.1 to obtain a general structure description for this nonabelian tensor square.

Corollary 1.1. Let G = Nn,c be the free nilpotent group of class c and rank n > 2. Then

G I8i G ~ Nn ,c+1 x F;b,

where g = ISn,c \ Sn,c-21 + (n!I).

2. The Derived Subgroup of a Free Nilpotent Group

In this section we first prove Theorem 1.1 and then use a result of Gaglione and Spellman7 to find a formula for the number ISm,w \ Sm,11 of simple left normed commutators in Cm,w of weight at least 2. From this formula we derive explicit expressions for s, the minimal number of generators for

Nn,w, and for ISm,w \ Sm,w-Il. The proof of Theorem 1.1 uses the following two theorems of Moran. I

Theorem 2.1 (Theorem 1.5, Ref. 1). Every abelian subgroup of a free

nilpotent group is free abelian.

Theorem 2.2 (Theorem 3.1, Ref. 1). Let B be a subgroup of a free

nilpotent group Nn,c of class c ~ 1 and rank n ~ 1. Then B is generated by

a set of c subgroups

where

48

(i) for k = 1,3, ... , c, the subgroup Bk is a free nilpotent group of class

LfJ; (ii) for n = 2, that is when Nn,e is 2-generated, the subgroup B2 is infinite

cyclic; otherwise the subgroup B2 is free nilpotent of class L ~ J ; (iii) for i + j ~ c, the subgroup [Bi' B j ] is contained in the subgroup

(Bi+j,'" ,Be); (iv) for i + j > c, the subgroup [Bi' B j ] is trivial; and (v) for k = 1,2, ... ,c - 1, the quotient group

is a free abelian group freely generated by the images of the free gener­ators of Bk in the quotient group.

It can be possible for a subgroup B of Nn,e that one or more of the Bi in Theorem 2.2 might have rank 0. In such cases we treat these subgroups as trivial.

Applying Theorem 2.2 to the subgroup Dn,e of Nn,e the subgroups Bl, ... , Be are constructed as follows. Set Bl to be a group of rank 0, as there are no basic commutators of weight 1 in Dn,e, and set

for k = 2, ... ,c. It follows that Dn,e is generated by Cn,e \ Cn,l. This fact can also be obtained by the Hall Basis Theorem.

We now prove Theorem 1.1.

Proof of Theorem 1.1. Let Nn,e be a free nilpotent group of class c ~ 1 and rank n ~ 1. If n = 1 or c = 1 then Nn,e is abelian and its derived subgroup Dn,e is trivial.

If c = 2 then Dn,e is abelian and hence free abelian by Theorem 2.1. The M(n, 2) commutators of Cn of weight 2 are independent and generate Dn,e. Hence Dn,e is free abelian of rank M(n, 2) = G).

Suppose c > 2 and n > 1. Let B l , ... , Be be the subgroups of Dn,e constructed above. The abelian subgroups Be and B e- l are free abelian by Theorem 2.1 and both are contained in the center of Dn,e. The Hall Basis Theorem states that Cn,e \ Cn,e-l is an independent generating set for Be. By property (v) of Theorem 2.2 the quotient (Be-l,Bc)/(Be) is freely generated by the images of Cn,e-l \ Cn,e-2. However, since B e- l is also central, (Be- l , Be) = B e- l X Be with rank ICn,e \ Cn,e-21. Let A be the subgroup of Dn,e generated by the basis elements Sn,e \Sn,e-2 of (Be- l , Be).

49

Let N be the subgroup of Vn,e generated by the set (Cn,e \ Cn,l) \ (Sn,e \ Sn,e-2). By Theorem 2.2 this subgroup is generated by subgroups B I ,.··, B~_l' B~ that satisfy properties (i)-(v). We define this sequence of subgroups as before, replacing the subgroups Be and B e- l with

B~ = ((Cn,e \ Cn,e-l) \ (Sn,e \ Sn,e-l)) and

B~_l = ((Cn,e-l \ Cn,e-2) \ (Sn,e-l \ Sn,e-2)).

It follows from A being central in Vn,e and from the Hall Basis Theorem that any element of Vn,e can be written as xy, where x E Nand YEA.

Hence Vn,e = N A. The subgroup N is normal in Vn,e since A is central and Vn,e = N A.

Any element 9 E N n A would have to be written both as a product of powers of the generators of N and as a product of powers of simple commutator generators of A. This is only possible if 9 = 1. Hence NnA = 1. Since both A and N are normal in Vn,e, it follows that Vn,e = N x A.

To show that N is isomorphic to Nn,e we define a new sequence of subgroups of N. For i = 2, ... , C - 2 we define

Set N* = (S2,"" Se-2). The generators of the subgroups Si for i = 2, ... ,C - 2 'are independent and cannot be expressed as products of pow­ers of the other generators. This holds since the generators are simple left normed commutators and no generator of N* has weight 1.

Using the bijection f3 the generators of K are mapped to the generators of N*. The groups K and N* both have the same nilpotency class, either l c/2 J if n > 2 or l c/2 J - 1 if n = 2. Hence the mapping of generators induces a homomorphism ¢ : K ---. N*. Since the Si are subgroups of the Bi then lSi, Sj] is trivial if i + j > c. Therefore the commutator [ki' kj ] is in the kernel of ¢ whenever the sum of the weights of the commutators f3(i) and ;3(j) is larger than c. Hence R ~ ker(¢). On the other hand, N* by construction does not introduce relations other than those found in Theorem 2.2. Hence R = ker(¢), and N* ~ K/R = Nn,e.

We complete the proof by showing that N* = N. If Ci is a generator of N that is not a simple left normed commutator then Ci = [cq , cp ], where Wq > 1 and wp > 1. If cq and cp are simple commutators then Ci is a product of the generators of N*. If either cq or cp is not a simple commutator then we repeat the process and determine that all generators of N are products of simple commutators that generate N*. 0

50

The following result from Gaglione and Spellman 7 enables us to provide precise values for the rank s of K and for f in Theorem 1.1 in terms of n and c.

Proposition 2.1. Let m and w be positive integers larger than 1. Then the value of ISm,w \ Sm,w-ll, the number of simple left normed commutators of Cm,w of weight exactly w, is

We derive an immediate consequence.

Corollary 2.1. Let m and w be positive integers greater than 1. Then the

value of ISm,w \ Sm,w-21 is

(m+w-2) ((W-1)+ W(W-2)).

w m+w-2

Proof. By Proposition 2.1,

ISm,w \ Sm,w-21 = ISm,w-l \ Sm,w-21 + ISm,w \ Sm,w-ll

= (w _ 2) (m; ~; 3) + (w -1) (m +: -2)

= (w _ 2) (m + w - 3)! + (w _ 1) (m + w - 2) (w - l)!(m - 2)! w

= (m + w - 2) ((W _ 1) + w(w - 2) ) . w m+w-2 D

Using Proposition 2.1 we may also determine a formula for the number ISn,w \ Sn,ll of simple left normed commutators of Cn,c of weight at least 2.

Proposition 2.2. Let w be a positive integer greater than 1. Let S:" = Sn,w \ Sn,l be the set of simple left normed commutators of Cn of weights 2, ... ,w. Then

IS* I = ~ IS . \ S . 1= (n + w - l)!(wn - w - n) + w!n!

w ~ n,) n,)-l , , . j=2 w.n.

Proof. The proof is by induction on w for each fixed value of n. For w = 2, the right side gives

(n + l)!(n - 2) + 2n! 2n!

(n + l)(n - 2) + 2

2

n(n -1) 2

which is equal to ISn,2 \ Sn,ll = G). Suppose that the formula holds for w = k 2: 2, that is, that

k

IS*I=""'IS .\8 . 1= (n+k-1)!(kn-k-n)+k!n! k ~ n,) n,)-l k' , .

j=2 .n.

Then, by the inductive hypothesis and Proposition 2.1,

18k+11 = 18kl + 18n,k+1 \ 8n ,kl

= (n + k - l)!(kn - k - n) + kin! + k(n + k - 1) kin! k + 1

(n + k - l)!(kn - k - n) + kin! k (n + k - I)! = +

kin! (k + l)!(n - 2)!

(k + l)((n + k - l)!(kn - k - n) + kin!) kn(n - l)(n + k - I)! = + ---''----'--'----'-

(k + l)!n! (k + l)!n!

(n + k - l)!(n + k)(nk - k - 1) + (k + l)!n!

(k + l)!n!

(n + k)!(n(k + 1) - (k + 1) - n) + (k + l)!n!

(k + l)!n!

51

which shows that the formula holds for w = k + 1, and thus the formula holds for all integers w 2: 2. 0

In particular, when w = c - 2,

* (n + c - 3)!((c - 2)n - (c - 2) - n) + (c - 2)!n! IS c-21 = (c - 2)!n!

(n + c - 3)!(cn - 3n - c + 2) + (c - 2)!n!

(c - 2)!n!

which is the value of the rank s of /C. We thus obtain the following refine­ment of the statement of our main result.

Theorem 2.3. Let Nn,c be the free nilpotent group of class c 2: 1 and rank

n> 2. Then

where

f=(n+c-2) ((C-1)+ C(C-2») c n+c-2

and Nn.,c has a minimal cardinality generating set with

(n + c - 3)!(cn - 3n - c + 2) + (c - 2)!n! s=

(c - 2)!n!

generators.

52

3. Application

In this section we apply Theorem 1.1 to describe the structure of the non­abelian tensor squares of the free nilpotent groups of finite rank.

Let G be any group. Then the group G 181 G generated by the symbols 9 181 h, where g, hE G, subject to the relations

for all g, h, and k in G, where Xy = xyx- 1 for x, y E G, is called the nonabelian tensor square of G. Let \7( G) be the subgroup of GI8IG generated by the set {g 181 gig E G}. The group \7(G) is a central subgroup of G 181 G (Brown and Loday4). The factor group GI8IGI\7(G) is called the nonabelian exterior square of G, denoted by GAG. For elements 9 and h in G, the coset (g 181 h)\7(G) is denoted 9 A h. Hence G 181 G is a central extension of GAG by \7 (G) and we have the short exact sequence

1 ----7 \7(G) a ----7 GAG ----7 1.

The following theorem from Blyth, et al. 3 provides the basic structure for the nonabelian tensor square of a free nilpotent group of finite rank.

Theorem 3.1. Let G = Nn,c be the free nilpotent group of class c and rank

n> 1. Then

G 181 G ~ f(GIG') x GAG,

where f( GIG') is the Whitehead quadratic functor defined by Whitehead. 8

Since the abelianization of Nn,c is a free abelian group of rank n, the

group f(N~~) is isomorphic to F(;;t') (WhiteheadB). It follows from Corol-

lary 2 of Brown et al.9 that 'Dn,c+1 is isomorphic to Nn,c A Nn,c' Putting these facts together we obtain Corollary 1.7 of Blyth et al.,3 which we now state.

Corollary 3.1. Let G = Nn,c be the free nilpotent group of class c and rank n > 1. Then

The following theorem combines our detailed description of 'Dn,c+l from Theorem 2.3 with Corollary 3.1.

53

Theorem 3.2. Let G = Nn,c be the free nilpotent group of class c and rank n> 2. Then

where

g= (n+c-l) (c+ (C+l)(C-l)) + (n+l). c+l n+c-l 2

and Nn,c+l has a minimal cardinality generating set with

generators.

(n + c - 2)! (( c + 1) n - 3n - c + 3) + (c - 1) !n! (c - l)!n!

Acknowledgements

The authors thank the Institute for Global Enterprise in Indiana for its financial support of this research. The first and second authors thank the University of Evansville for its hospitality while visiting there. The sec­ond author thanks the Ministry of Science of Slovenia for supporting his postdoctoral leave to visit the University of Evansville.

References

1. S. Moran, A subgroup theorem for free nilpotent groups, Trans. Amer. Math. Soc. 103, pp. 495-515 (1962).

2. N. P. Gol'dina, Free nilpotent groups, Dokl. Akad. Nauk SSSR (N.S.) 111, pp. 528-530 (1956).

3. R. D. Blyth, P. Moravec and R. F. Morse, On the nonabelian tensor squares of free nilpotent groups of finite rank, in Computational Group Theory and the Theory of Groups, eds. L.-C. Kappe, A. Magidin and R. F. Morse (American Mathematical Society, Providence, RI, 2008)

4. R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26, pp. 311-335 (1987), With an appendix by M. Zisman.

5. R. K. Dennis, In Search of New "Homology" Functors having a Close Rela­tionship to K-theory, Unpublished preprint.

6. C. Miller, The second homology group of a group; relations among commuta­tors, Proc. Amer. Math. Soc. 3, pp. 588-595 (1952).

7. A. M. Gaglione and D. Spellman, Extending Witt's formula to free abelian by nilpotent groups, J. Algebra 126, pp. 170-175 (1989).

8. J. H. C. Whitehead, A certain exact sequence, Ann. of Math. (2) 52, pp. 51-110 (1950).

9. R. Brown, D. L. Johnson and E. F. Robertson, Some computations of non­abelian tensor products of groups, J. Algebra 111, pp. 177-202 (1987).

A RECURRENCE RELATION FOR THE NUMBER OF FREE SUBGROUPS IN FREE PRODUCTS

OF CYCLIC GROUPS

T. CAMPS

Fachbereich Mathematik, Universitiit Dortmund, Vogelpothsweg 87, 44137 Dortmund, Germany

E-mail: thorsten.camps@mathematik.uni-dortmund.de www.mathematik.uni-dortmund.de

M.DORFER

Am Beutenbach 24 71254 Ditzingen, Germany

E-mail: mb.doerfer@t-online.de

G.ROSENBERGER

Fachbereich Mathematik, Universitiit Dortmund, Vogelpothsweg 87, 44137 Dortmund, Germany

E-mail: GerhardRosenberger@mathematik.uni-dortmund.de www.mathematik.uni-dortmund.de

Dedicated to A. M. Gaglione on the occasion of his 60th birthday.

In this note we consider a free product G of finitely many cyclic groups of finite or infinite order and develop an explicit and straightforward recurrence formula for the number of free subgroups of G which includes only the given group theoretic data as the number of the free product factors and the orders of the given cyclic groups.

Keywords: Free Products, free subgroups, recurrence formulas for free sub­groups. Mathematical classification: 20E06, 20E07, 20F05, 20HlO.

54

55

1. Introduction

We consider free products of finitely many finite or infinite cyclic groups of the following form:

G = Cr1 * Cr2 * ... * Crd * Coo * ... * Coo '-..-'

u factors

(1)

where d and u are nonnegative integers with d + u ::::: 2, 7"i E 1'1, 7"i ::::: 2

(i E {1,2, ... ,d}), 7"1 ::::: 7"2 ::::: ... ::::: 7"d. Moreover we define t:= 1 if d = 0

and t := lcm (7"1,7"2, .•. , 7"d) if d ::::: 1. In this case let t := 7"iSi for all i E {1, 2, ... , d} .

For these groups, Stothers [10] gave a recurrence relation for Fc (n), the number of free subgroups of G of index n, namely

Fc (n) = 0 if n '1= 0 (mod t)

and for kEN,

1 ~ k-l 1 ~ Fc (tk) = (tk _ 1)! h (G, Stk) - L (tk _ tj)! h (G, Stk-tj) Fc (tj) (2)

)=1

where

h (G S ) = (rrd

(tk)! ) ((tk)!)U , tk ( .k)' Si k

i=l S, ·7"i

(3)

is the number of homomorphisms from G to the symmetric group on a set M of tk elements satisfying the condition that the preimage of the stabilizers of the elements of M are free or trivial.

The disadvantage of this recurrence relation is that one has to determine the number of homomorphisms from G into symmetric groups. The advan­tage of this recurrence relation is that it is easy to extend it analogously to

a general recurrence relation for arbitrary free products H = HI * ... * H n ,

n ::::: 2, which is difficult to calculate explicitely but from which one gets a

nice asymptotic behaviour for the number of subgroups in H of a given in­

dex. (see for instance [8] and [9]) An earlier general version for the number of subgroups of finite index in H is given by Dey [1]. This version was used to count the subgroups of finite index for the modular group C2 * C3 and more generally for the Hecke groups which are free products C 2 * Cq of two cyclic groups, one of order 2 and one of order q ::::: 3 (see for instance [2],

[3], [4], [5] and [7]). Apart from formula (2), Stothers [10] also studied a second recurrence

relation for the number of free subgroups of G of index n and gave it

56

explicitly in the simplest cases Coo * Coo, C2 * Coo, C3 * C3 , C2 * C3 , C2 * C4 ,

C2 * C2 * C2 and C2 * C2 • In this paper we present the formula for all free products of type (1), thus generalizing the results of Stothers, and furthermore of Wohlfahrt [11] for the modular group C2 * C3 and of Kern­Isberner [6] for some Hecke groups C2 * Cq , q 2': 3.

Before we can state the main result, we need a couple of preliminary observations.

2. Preliminaries

For a group G of type (1) let

h (G, Stn) O'n : = () , (n E N), tn.

0'0 := 1,

00

00

f:= LO'nzn , n=O

an := Fc (tn) (n EN), g:= L an+1zn n=O

(I, 9 formal power series). With these definitions, (2) is equivalent to the differential equation

f' g=t-f

(4)

which was used by Wohlfahrt in [11] to derive the recurrence relation for the number of subgroups in the modular group.

Lemma 2.1. Let n E No be arbitrary but fixed. Then we have

k-1 nk-n(n-1) ... (n-k+1)=Ln1dk,1 (kEN) (5)

1=1 with

dk,l = (_1)k-I-1 (k - I)! (L 1) (6) ip1" 'ipl-1

('1'1 , ... ,'PI-ll E{1,2, .. . ,k-1} 1-1 1 ::0'1'1 < ... <'1'1-1 ::ok-1

and the sum in (5) being defined as 1 if l = 1.

Proof. Observing that dk,k = 1, we can see that (5) is equivalent to

k

n(n-1) ... (n-k+1)=-Ln1dk,1 (kEN) 1=1

57

which can be proved easily by induction on k. o

Lemma 2.2. Let n E No be arbitrary but fixed. Then

h h

LKmnm = L n (n-1) ... (n-m+1) m=1 m=1

for all hEN and all real Km (m E {1,2, ... ,h}} where the di/",i/"+l are defined by (6).

Proof. The case n = 0 is trivial, so let n E N. For h = 1, the result is obvious, and we calculate

,+1 h

LKmnm= Ln(n-1) ... (n+m+1) m=1 m=1

h

+Kh+1n (n - 1)··· (n - h) + Kh+1 L n mdh+1,m m=1

h+1 h = Ln(n-1) ... (n-m+1)Km+ Ln(n-1) ... (n-m+1)

m=1 m=1

(7)

+ K h+1 dh+1,m (8)

+ J.1;+1 (Kh+1 dh+1,j j~1 L dil,i2di2,i3" .dik_l,ik)] (il , ... ,ik )EN k

j=il> ... >ik=m

58

using the induction hypothesis twice and Lemma 2.1. Furthermore

h

L

k=2 (i" ... ,ik)ENk

h+1=i,> ... >ik=m

If we put this result into (8), we are done. o

Applying the differential equation (4) yields

Lemma 2.3. Let mEN, then

with natural numbers el,', ... ,lk which can be calculated recursively by the fol­lowing formulas for m 2: 2 using eb = 1, defining j1, j2, ... , jc-1 for a positive integer c to be those indices in {I, 2, ... , k - I} for which I ji > I ji +1

(i E {1,2, ... ,c-l}), and letting jc:= k:

( ii)

( iii)

c-1

"'"' z·em

-1 'f I 0 ~ '1" ... ,lji-

"lji- 1,lji+

" ... ,lk ~ k = ,

i=l

Zi being the number of coordinates in (lr, ... , Iji-1, Iji - 1, Iji+1, ... , lk) which are equal to Iji -1 (i E {1,2, ... ,c}),

59

Proof. We will not give the proof in detail but mention the most important ideas. Step 1:

f (m) m 1 t- - '" _sm (' (m-k)) f - ~tk-I k g,g, ... ,g ,

k=1

(m E N)

where the S'k(g,g', ... ,g(m-k)) are defined recursively for m > 2 and k E {I, 2, ... , m} by

m I (m-k)._ m-I I (m-I-k) Sk g,g, ... ,g .- Sk g,g, ... ,g ( ) ( ( )) '

Sm-I (' (m-k)) +g k-I g,g, ... ,g

with Si (g) := 9 and

Sm-I (' (m-I)) ._ 0 _. sm-I (' (m-I-m)) o g, 9 , ... , 9 .- -. m g, 9 , ... , 9

for all m ;::: 2.

Proof of step 1: Induction on musing

Step 2: Every summand in S'k (g, g', ... , g(m-k)) (m E N, k E

{1,2, ... ,m}) is of type

where a E N, li E {O, 1, ... , m - k} (i E {l, 2, ... , k}) and h +l2 + ... +lk =

m-k.

Proof of step 2: Induction on m. We only look at the induction step from m - 1 to m (m ;::: 2). The cases k = 1 and k = m are trivial because

S m (' (m-I)) sm-I (' (m-2)) I I g,g, ... ,g = I g,g, ... ,g

(then use the induction hypothesis) and S: (g) = gm.

For k E {2,3, ... ,m -I},

m I (m-k) m-I I (m-I-k) Sk (g,g, ... ,g ) = (Sk (g,g, ... ,g )) '

Sm-I (' (m-k)) +g k-I g,g, ... ,g ,

60

so that every summand of Sf (g, g', ... , g(m-k)) is either one of

(sm-l (' (m-l-k)))' f Sm-l (' (m-k)) d th k g, 9 , ... , 9 or 0 9 k-l g, 9 , ... , 9 an ose are known by the induction hypothesis.

Step 3: Let (h, ... , lk) E N~ with II ::::: l2 ::::: ... ::::: lk = m - k where mEN and k E {I,2, ... ,m}. Then Sf (g,g', ... ,g(m-k)) contains a summand which is equal to the product of g(lI) g(l2) ... g(lk) and a suitable positive

integer.

Proof of step 3: Induction on m. The case m = 1 is trivial. The step from m - 1 to m (m ::::: 2):

Let k E {I,2,oo.,m}, (l1,oo.,lk) E N~ with h::::: l2 > h + l2 + ... + lk = m - k. a) Let lk = O. Thus k ::::: 2. Belonging to (ll, ... , lk) we have the product

where (h, ... ,lk-l) is a (k - I)-tuple with II ::::: l2 ::::: ... ::::: lk-l and h + l2 + ... + lk-l = (m - 1) - (k - 1), so that there is (by induction hypothesis) a summand with respect to (l1, ... , lk-d in S~11 (g,g', ... ,g(m-k)) with the desired property. Then use the re-

I t · f sm (' (m-k)) currence re a IOn or kg, 9 , ... , 9 .

b) Let lk =I- O. Thus k :s: m - l. Belonging to (h, ... , lk) we have the product

where (h, ... , lk-l, lk - 1) is a k-tuple with II +l2 + .. . +lk-l +lk -1 =

(m - l)-k and II ::::: l2 ::::: ... ::::: lk-l ::::: lk-I ::::: 0, so that there is (by the induction hypothesis) a summand with respect to (h, ... , h-l, lk - 1) in S;;-1 (g,g', ... ,g(m-l-k)) with the desired property. Then use the recurrence relation for Sf (g, g', ... , g(m-k))

Step 2 and 3 together yield

sm (' (m-k)) k g,g, ... ,g =

(11 , ... ,lk)EN~ h +l2+ ... +lk=m-k2:l12:l22: ... 2:lk

61

with coefficients eZ:, ... ,lk E N (m E N, k E {I, 2, ... , m}) still to be deter­mined, and we finally see

(m E N) by step 1. It remains to prove the recurrence relation for the coefficients.

Obviously e6 = 1 because 9 = Sf (g) = e6g(0).

Now let m ;::: 2 and k E {I, 2, ... , m}, moreover define j1, j2, ... , jc-1 for a positive integer c to be those indices in {I, 2, ... , k - I} for which l ji > l ji +1

(i E {1,2, ... ,c-1}), and finally let jc:= k. (i) follows immediately from the fact, that eZ:, ... h g (lll ... g(lk) is a sum­

mand of Sf (g,g', ... ,g(m-k)), and the recurrence relation from step 1 provides us with the key information to realize (i).

(iii) will be proved by induction on m. We will refer to the induction hypothesis by "IH". The case 6 = k being clear, we can assume (h, ... ,lk) E N~ with h ;::: l2 ;:::

... ;::: l8 > 0, lH1 = lH2 = ... = lk = 0 and h + l2 + ... + lk = m - k for a 6 E {O, 1, ... , k - I}. The case m = 2 is trivial. The step from m - 1 to m (m ;::: 3): If 6 = 0 then h ... = lk = 0, so

k = m and eml 1 = 1 = (k':O) . 1. 1"", k

If 6 E {1,2, ... ,k -I} (=} c;::: 2) then jc-1 = 6 and

{I if l8 > 1,

Zc-1 = k - 6 + 1 if l8 = 1.

It follows

IH m - 1 m-1-(k-1)H m -1 m-1-kH c-2 ( ) = e + Zi e... (k -1- 6) L" ... ,lo t; k -6 l" ... ,IJi-

"IJi- 1,11i+ 1 , ... '/0

m-1 +Zc-1 eL, , ... ,10-1 ,10 -1,0, ... ,0'

(10)

Moreover we have (in case of l8 > 1)

m-1 IH (m -1) m-1-k+8 Zc-1 e L" ... ,10_1,10-1,0, ... ,0 = k _ 6 eL" ... ,10_1,10-1

62

and (in case of lo = 1)

m-1 !!i (k 5 + 1) ( m - 1 ) m-1-k+o-1 zc-1el l , ... ,10_1,l0-I,O, ... ,o - - k - 5 + 1 ell, ... ,lo_l

(m - 1) (m -1 - k + 5) m-1-k+o-o+o-l k - 5 5 - (5 - 1) ell, ... ,lo_l

IH (m -1) m-l-k+o k - 5 ell, ... ,lo_l,O·

So (10) together with a further application of (i) gives

m ( m -1 ) m-k+o (m - 1) m-k+o e - e + e -h, ... ,lk - k - 1 - 5 h, ... ,lo k - 5 11, ... ,1 0 -

(ii): This is the lengthiest part with numerous calculations. The case m = 2 is clear. The step from m - 1 to m (m ~ 3): We first consider the case lk = 0, that is it ~ ... ~ lo > 0 and lHI = ... = lk = 0 for a 5 E {O, 1, ... , k - I}. Then of course k ~ 2 and (because the formula is correct for 5 = 0) we can assume 5 ~ 1, thus c ~ 2. By applying (iii), the induction hypothesis and once more (iii) it follows

c-1 () m m-1 m-l-1·. L m - k + 5 l· eh, ... ,I/~l,lji+l, ... ,lo,O, ... ,o i=1 j,

c-1 ( ) m -1 m-l-1' L l.. eh,···,ljiJ~l,lji+l, ... ,lo,O, ... ,o i=1 j,

c-l ( ) k - 5 m - 1 m-l-1 + e J, • m - k + 5 L l· h, ... ,lji-l,lji+l,"',lo,o, ... ,o

i=l j,

(11)

Because of jc = k and lk = 0 we get

(m - 1) m (iii) ( m - 1 ) m-k+o

lje eh, ... ,lk_l = k - 1- 5 ell, ... ,lo

= ~ (( m -1 ) (m - k + 6 -1) k-1-O' 1.

i=l ].

k - 6 c-1 (m -1) m-1-1 .. == e Jt m - k + 6 L 1· 11, ... ,lji- 1,lji+ 1, ... ,18,0, ... ,0 ,

i=l J'/.

so the asserted formula can be derived from (11). Let now lk =1= O. Omitting the trivial case k = 1, we suppose k ~ 2.

If c = 1 then h = l2 = ... = lk and Zl = 1, so

In the following, we examine the case c ~ 2.

We introduce here the Notation: e~,~l'~klll~i := e~,~~,~:j~1,lji+1, ... ,lk·

We consider several cases of values for i and ji. i = 1 andji ~ 2 or i E {2,3, ... ,c-1}:

Case 1: lj.+1 < lji - l. a) lji = lji- 1 . Then

63

64

b) Iji < Iji- 1 . Then

Case 2: Iji+1 = Iji - 1.

a) Iji = lji-l' Then:

m-l eLI , ... ,lji -1 ,lji -1,lji+1 , ... ,lk

i = 1 and]1 = 1: Case 1: h - 1 = Z2. Then:

Case 2: II - 1 > Z2. Then:

er';=t,12, ... ,lk ~ ~ (mz: 2)e;,;~;,~:~.~.'lklljr + (~= De~,~.I,lkh. i = c:

Case 1: Zk-l = Ik. Then:

65

Case 2: lk-I > lk. Then:

m-I IH m - 2 m-2-1· m - 2 m-I-1k c-I () () e = e Jr + e . It, ... ,lk-l,h-I L l· 11, ... ,lk_l,lk-II1jr l _ 1 It, ... ,h-l r=1 )r k

Now define integers lo and lk+1 such that lo - 1 > hand lHI < lk - 1. Then the four investigated special cases for i = 1, ji = 1 and i = c (in the above order) may uniquely be assigned to the formerly discussed cases 2.b), 1.b), La), 1.b), and by

we get

M I := {i 11:::; i:::; C,lji+1 < lji -1,lji = lji-d,

M 2 := {i 11:::; i:::; C,lji+1 < lji -1,lji < lji-d,

M3 := {i 11:::; i:::; C,lji+1 = lji -1,lji = lji-d,

M4 := {i 11:::; i:::; C,lji+1 = lji -1,lji < lji-d

66

i=l lji- 1 =lji

+ L i=l

lji-1>lji

+ L i=l

iji -l>iji+l

+ L i=l

lji- 1 =lji

+ L i=l

iji -1>iji+1

+ L r=l

ljr- 1 =ljr

c

+ L r=l

ijr -l>ijr+l

On the other hand

Let r 2': 2. Case 1: ljr-l - 1 = ljr, that is ljr_l = ljr-l.

(12)

67

em-l-ljr it , ... ,ljr-1 ,ljr+l '" .,lk

Case 2: ljr-1 = ljr'

a) ljr_l - 1 = ljr' Then:

Let r = 1. a) jl = 1. Then:

b) jl 2': 2. Then:

which can uniquely be assigned to the cases 1.b) and 2.b) using the con­vention for la and moreover jo := O. So (13) is equal to

68

c

1'=2 ljr_l-1=ljr

c

+ L 1'=1

Ijr -1>ljr+1

c-1

+ L 1'=1

Ijr-1=ljr+1

which is equal to (12). This concludes the proof of Lemma 2.3.

3. The Main Results

Theorem 3.1. For a group of type (1) let an = Fc (tn), then

and

o

(14)

(15)

(n ~ 1) where hand Lm (m E {1, 2, ... , h}) are given thus:

d

h := t (u + d - 1) - LSi + 1, i=1

In the last formula

with

69

T:= ( 1, ... ,1,2, ... ,2 , ... ,t-1, ... ,t-1,t, ... ,t) =:(Tl,T2, ... ,Th), ~ ~ '-v-' "-v--"

u+d-l-w1 u+d-l-w2 u+d-l-Wt_1 u times

wI" is the number of sets Vrl! ... , Vrd of positive multiples of '1'1, ... , 'I'd

which contain fL (fL E {1, 2, ... t - 1}),

1 bT,w:= L T ... T (wE{1,2, ... ,h})

( ) J1 Jw

Ti1,···,Tjw

l:Sh < ... <jw:S h

Moreover, the di",i,,+1 and el,', ... ,lk are defined as in (6) on page 56 and (9) on page 58 respectively.

Proof. By (3) (see page 55), we have

so

or equivalently

an = h (G, Stn)

(tn)!

t n (tn + It+ d-

1

1=1 a n +l = d s, an

n n (tn + jri) i=1 j=1

t-l n (tn + It+ d-

1

t (n + 1) an+! = (tn + tt 1;1 s,-1 an (if d ~ 1) (16) n n (tn + jri) i=1 j=1

70

and

for all nonnegative integers n. In (16), the right side can be written as a polynomial in n. For d 2': 1 and i E {I, 2, ... , d} let Vri be the set of all positive multiples of ri and let Wz be the number of Vri containing l (l E {I, 2, ... , t - I}). Then (16) becomes

t (n + 1) an+! = (tn + tt (g (tn + It+d- 1- W1 ) an. (n E No) (17)

(Using t = 1 if d = 0, the formula in case of d = 0 is contained therein.) The product

(

t-1 ) (tn + tt g (tn + l)u+d-1-W

l

in (17) consists of

d

t (u + d - 1) - LSi + 1 =: h(2': 1) i=l

factors, so we can expand the right side of (17) as a polynomial in n of degree 2': 1. For this purpose we define

T:= ( 1,,,.,1,2,,,.,2 ,,,.,t-1,,,.,t-1,t,,,.,t) =: (T1,T2,,,.,Th), '----v-" '----v-" ~ ~

u+d-1-W1 u+d-1-w2 u+d-1-Wt_1 u times

and we get

where

bT,m := 1

(mE{l,2,,,.,h})

(Tj1 , ... ,TjTn) T- .. 'T-J1 J~

l::;ji < ... <j~::;h

and bT,o := 1. By virtue of (7) (see page 57) and

m ((t - 1)!)u+d-1 t U

Km = t d bT,m (m E {O, 1,,,., h}) IT rti

-1 (Si - I)!

i=l

we get

t (n + 1) an+!

h

=Koan + L n (n-1) ... (n-m+1)a n

m=l

for all n E No which is equivalent to the differential equation

h

tf' = Ko! + L zm !(m) m=l

or after division by ! to

In terms of the formal power series g, the last equation reads

by (4) (page 56) and (9) (page 58). Observing that

comparison of the coefficients yields the desired result.

71

o

72

4. Examples

Formula (15) contains (h - l)-fold sums, so easy expressions can only be

expected if hE {1, 2}. h = 1 is exclusively possible for d ~ 2, namely for the group G = C2 *C2 ,

and (14), (15) give

a1 = 1,

an+1 = L 1an = an = 1, (n ~ 1)

as was shown by Stothers in [10]. The case h = 2:

For d = a we have h = 2 if and only if u = 2, that is G = Coo * Coo. For d = 1 we have h = 2 if and only ift = 2 and u = 1, that is G = C2 *Coo · For d ~ 2 we have h = 2 in exactly the following cases:

G =C3 * C3 ,

G =C2 * C3 , G = C2 * C4 ,

G =C2 * C2 * C2.

(d = 2,81 = 82)

(d=2,8l=82+ 1)

(d = 3)

For all of these groups, formula (15) is (because of eb = 1, 2I = 1, e6,o = 1) of the form

L2 n-l

an+l = Llan + L2 (n - 1) an + t L ak+1 an-k-1 (n ~ 1) k=O

where

with

Kl = tKobT,l, K2 = t 2 K obT,2 = t 2.

We calculate the following data table:

G Ko T bT,l bT,2 Kl K2 L1 L2

Coo * Coo 1 {1,1} 2 1 2 1 3 1

C2 * Coo 2 {1,2} 3 1 6 4 5 2 "2 "2 C3 * C3 2 {1,2} 3 1 9 9 6 = 2t 3=t "2 "2 C2 * C3 5 {1,5} 6 1 36 36 12 = 2t 6=t 5 5 C2 * C4 3 {1,3} 4 1 16 16 8 = 2t 4=t "3 "3 C2 * C2 * C2 1 {1,1} 2 1 4 4 4 = 2t 2=t

so that we finally have for G = Coo * Coo:

n-l

an+l = (n + 2) an + L akan-k (n;:::: 1) k=l

and for G = C2 * Coo:

n-l

an+l = (2n + 3) an + L akan-k· (n;:::: 1) k=l

In all other cases:

n-l

an+l = t (n + 1) an + L akan-k· (n;:::: 1) k=l

73

These are the special cases considered by Stothers in [10]. (The "-" in front of the sum has to be corrected there.) The formula for G = C2 * C3 may also be found in [11]. If we take the group G = C2 * C6 and use (14) and (15), we deduce the result from [6].

References

1. 1. M. S. Dey, Schreier Systems in Free Products; Proc. Glasgow Math. 7 (1965), 61-79.

2. B. Fine, D. Spellman, Counting Subgroups of the Hecke Groups; Int. J. of Algebra and Computation 3 (1993), 43-49.

3. M. Grady, Counting the Subgroups of an Infinite Group; JCMCC 20 (1996), 89-96.

4. M. Grady, M. Newman, Counting Subgroups of Given Index in Hecke Groups; Contemporary Math. 143 (1993), 431-436.

5. W. Imrich, On the Number of Sugroups of a given Index in SL2 (/Z); Archiv d. Math. 31 (1978), 224-231.

6. G. Kern-Isberner, Rekursionsformeln fur die Anzahl von Normalteilern in freien Produkten zyklischer Gruppen; Dissertation, Dortmund, 1985.

7. Mong-Lung Lang, Chong-Hai Lim, Ser-Peow Tan, Subgroups of the Hecke Groups with Small Index; Lin. and Multil. Algebra 35 (1993), 75-77.

8. T. Muller, Counting Free Subgroups of Finite Index; Archiv d. Math. 59 (1992), 525-533.

74

9. T. Miiller, Subgroup Growth of Free Products; Invent. Math. 126 (1996),111-131.

10. W. W. Stothers, Free Subgroups of the Free Product of Cyclic Groups; Math. Camp. 32 (1978), 1274-1280.

11. K. Wohlfahrt, Uber einen Satz von Dey und die Modulgruppe; Arch. Math. 29 (1977), 455-457.

The Baumslag-Solitar Groups: A Solution for the Isomorphism Problem

Anthony E. Clement

Department of Mathematics, Brooklyn College, The City University of New York, 2900 Bedford Avenue Brooklyn, NY 11210, USA

E-mail: aclement@brooklyn.cuny.edu

Abstract: The object of this note is to give a concise proof of the following theorem: Let G = B(m,n) = (a, bi a-1bma = bn 1m # 0, n # 0, m, n E Z) and H = B(m',n') = (X,Yi x-1ym'x = yn'lm' # 0, n' # 0, m', n' E Z). Then G eo< H if and only if m = m' and n = n'.

Keywords: Baumslag-Solitar groups, isomorphism problem, semi-direct prod­ucts

1. Introduction

The theorem quoted in the abstract above solves, in principle, the isomor­phism problem for the class a = B(m,n) = (a, b; a-1bma = bn I m =I=- 0, n =I=-

0, m, n E Z) of the Baumslag-Solitar groups. D.l. Moldavanskii's proof [4] was rather different. The approach we take here is more direct and makes use of the fact that certain quotients of these groups can be expressed as semi-direct products of a very special kind. We capitalize on the inher­ent semi-direct product nature of B(m,n) and analyze inherited actions of the infinite cyclic group on some specific subgroups of B(m,n) and their quotients. We will prove the following:

Theorem 1.1. Let a = B(m,n) = (a, b; a-1bma = bn I m =I=- 0, n =I=-

0, m, n E Z) and let H = B(m',n') = (x,y; x-lym'x = yn' 1m' =I=- 0, n' =I=-

0, m', n' E Z). Then a ~ H if and only ifm = m' and n = n'.

2. Notation

The following notation is used throughout: 0(1)= [a, a] for the first derived group of a

75

76

G(2)= [G(l), G(1)l for the second derived group of G G = S )<I (a) for G written as a semi-direct product of S by (a) G ~'P H to mean G is isomorphic to H via a particular isomorphism r.p gpc(b) for the normal closure of bin G ((G) for the center of the group G r.p*IB~ to mean the restriction of r.p* to B~ (a) for the cyclic group generated by the element a

3. The proof of Theorem 1.1

We reserve the notation B(m,n) and B(m', n') for the Baumslag-Solitar groups (a, b; a-1bma = bn I m -I- 0, n -I- 0, m, n E 7L) and (x, y; x-lym' X = yn' I m' -I- 0, n' -I- 0, m', n' E 7L), respectively ([1]). We note that the reverse direction of Theorem 1 is obvious. Proving the forward direction will complete our task. Since Tietze transformations show that B(m,n) ~ B(-m,-n), without loss of generality, we will assume that n - m ;::: O.

We will need the following two lemmas whose proofs (see, e.g., [2]) can be easily deduced through some obvious observations and from Magnus' breakdown ([3]) of I-relator groups.

Lemma 3.1. Let G = B(m,n) with n - m ;::: 0 and H = B(m',n') with n' - m' ;::: O. If G ~ H, then n - m = n' - m'.

Lemma 3.2. Let G = B(m,n) with n - m ;::: O. If S/G(l) is the torsion subgroup ofG/G(1), then (1) S = {g E Gil E G(l) for some k > 0, k E 7L} = gPa(b); (2) S = ( ... , bi , ... (i E 7L); ... , bi = bf-l' ... (i E 7L)), where bi = a-ibai , (i E 7L) ; (3) G/S ~ 7L.

Now we turn to the proof of Theorem 1.1 in the forward direction. We first take care of the case n - m = O. Observe that (B(m,m) = gp(bm) and B(m,m)/gp(bm) ~ 7L * 7Lm. If

G ~ H, then n' = m' by Lemma 3.1. So since B (m, m) / gp( bm ) ~ 7L * 7Lm

and B (m' , m') / gp(bm') ~ 7L * 7Lm" m = m' and n = n' follows. So we can assume from now on that n - m > O.

Let T = {h E Hlhk E H(l) for some k > 0, k E Z}. Then, as in Lemma

3.2, we find T = gPH(Y) = ( .. ·,Yi, ... (i E 7L); ... ,Yi' = Y~l, ... (i E Z)) and H/T ~ 7L.

Now we are in position to formulate the following lemma whose proof relies on ideas similar to those for Lemma 3.1.

77

Lemma 3.3. Let a = B(m,n) with n - m > 0, H = B(m',n') with n'­m' > 0, and <p a particular isomorphism between a and H. Put PIS(l) and Q IT(l) for the torsion subgroups of S I S(1) and T IT(l), respectively. If

a ~'P H, then a I P ~'P* HI Q and SIP ~'Pu T I Q, where both <p* and <pH

are the isomorphisms induced by <po

We analyze how certain actions are inherited in the quotients of a. It is clear (cf. Lemma 3.2) that

0= S)(J (a),

where (a) is the infinite cyclic group on a, that SIS(1) = ( ... ,bi, ... (i E

Z); ... , bi = bi-l' ... (i E Z), ... , rbi, bj ] = 1, ... (i, j E Z)) and the action of (a) in a induces a similar action in a I S(1). Hence,

which implies that

Thus

alP ~ SIP)(J (aP).

Now we put L = alP, B" = SIP, a" = (aP). So L = B" )(J

(a"), where the subgroup B" = ( ... ,b~, ... (i E Z); ... ,b~m = b~~l, ... (i E

Z), ... ,[b~,bj] = 1, ... (i,j E Z)) is torsion-free abelian. A typical element of L has the form {a" r b'l l' > 0, l' E Z, b' E B"}. Moreover, the relation a"-lb,ma" = b'n holds in L. To see this, observe that (the subscripted ver-sion) all-lb~mall = b~n holds in Land b' has the form b' = b~~1 ... b~:l, where

ti = ±l. Since the b~'s commute, a"-lb,ma" = all-l(b~~1m ... b~~lm)all =

1I- 1b'E1 m II l-lb'E2ma" a"-1b'E1m II = b'E1nb'E2n b~Eln = (b'E1 b'El)n = a 21 a a 22 ... 21 a 21 22 ... 2£ 21 ... 21 '"-.".-''"-.".-' '"-.".-' b'n. Similarly, analyzing actions inherited in analogous quotients of

H = T)(J (x),

we see that T IT(l) = ( ... , Yi, ... (i E Z); ... , Yi' = Yi~l' ... (i E Z), ... , [Yi, Yj] = 1, ... (i,j E Z)), whence

which implies that

HIT(l) /QIT(l) = TIT(l) /QIT(l) )(J ((xT(l»)QIT(l»)

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and so

H/Q ~ T/Q )<l (xQ).

Letting M = H/Q, yH = T/Q,and x" = (xQ),now M = yU )<l (x") U ' , follows, where Y = ( ... , y~, ... (i E Z); ... , yr = y~'?:.1' ... (i E Z), ... , [y~, yjl =

1, ... (i,j E Z)) is torsion-free abelian. A typical element of M has the form {x"ky'l k E Z, y' E yU}, moreover, the relation x,,-1 y,m' x" = y,n' holds in

M. We now invoke Lemma 3.3 for L = G/P and M = H/Q. SO <p* : L ~M

induces an isomorphism rp : L/BU ~ M/YU defined by rp(a"BU) = <p*(a")YU. Since both L/BU and M/YU are infinite cyclic, rp is an isomor­phism. This implies that either <p*(a")YU = x"yU or <p*(a")YU = x,,-1yU.

It is clearly enough to consider the case <p* (a") yU = x"yU, so that x"-1<p*(a")YU = yU. Thus xl-1<p*(a") E yU and therefore X,,-1 <p* (a") = yU, for some yU E yU. Hence

<p*(a") = x"yU, where yH is uniquely determined by <p*(a"). Next we look at the consequence of the relation a"-1b'ma,, = b'n in L

under <p* : L ~M. <p*(a"-1b,ma") = <p*(b'n), which implies that <p*(a,,)-1<p*(b,m )<p*(a") = <p*(b'n), where <p*(b' ) = yl E yU. Thus we have (x" yU) -1 y'm (x" yU) = y'n, (since <p*1 B# = <pU and <pU : BU ~ yU) leading to yU-1x"-1ylmx"yU = yin, x,,-1 ylmx" = yUylnyU-1, and x,,-1 y,mx" = yin.

By taking m lth powers in x,,-1 ylmx" = yin, (x"-1 ylm x ,,)m' = (yln)m'. Since elements y' E yU commute,

l ' , x"- ylmm x" = (yln)m , and (x,,-1 ylm' x,,)m = (yln)m'. (*)

On the other hand we also know that x,,-1 ylm' x" = yIn' holds in M, so by taking mth powers in x,,-1 ylm' x" = yIn', we obtain (x"-1 ylm' x,,)m = (yln')m. However, by (*) this implies (yln)m' = (yln')m, ylnm' = yln'm. Thus

nml = n'm must hold, in addition to n - m = n l

- m l (Lemma 3.1). This implies that n2 - nm = nn' - nm', so n2 - nm = nn' - nlm, which is the same as n(n - m) = n'(n - m).

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Now if n - m =I- 0, n = n' follows. But since n = n' and nm' = nm, we get m=m'. This completes the entire proof of Theorem 1.1

Acknowledgments

Grateful thanks are offered to Gilbert Baumslag and Katalin Bencsath for extending their generous help during the writing of this paper.

References

1. Gilbert Baumslag, Donald Solitar, Some Two-generator One-relator Non­Hopfian Groups, Bull. Amer. Math. Soc. 68 (1962), 199-201.

2. Anthony E. Clement, On the Baumslag-Solitar Groups and Certain General­ized Free Products, Ph.D. Thesis, The Graduate Center, The City University of New York, New York, New York 10016, October (2006).

3. Wilhelm Magnus, Das Identitiitsproblem fur Gruppen mit einer definierenden Relation, Math. Ann. 106, (1932) 295-307.

4. D.1. Moldavanskii, Isomorphism of the Baumslag-Solitar Groups, Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, no. 12, Decem­ber (1991), 1684-1686.

Unification theorems in algebraic geometry

E.Daniyarova

Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 13 Pevtsova Street, 644099 Omsk, Russia

A.Myasnikov

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Montreal QC H3A 2K6, Canada

V.Remeslennikov

Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 13 Pevtsova Street, 644099 Omsk, Russia

Dedicated to Tony Gaglione on his Sixtieth Birthday

Abstract: In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A.

CONTENTS

1 Introduction

2 Preliminaries 2.1 Languages and Structures 2.2 Theories

3 Algebras 3.1 Congruences 3.2 Quasivarieties 3.3 Universal Closures 3.4 A-Algebras

4 Types, Zariski topology and coordinate algebras 4.1 Quantifier-free types and Zariski topology 4.2 Coordinate Algebras and free types

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4.3 Equationally Noetherian Algebras

5 Limit algebras 5.1 Direct systems of formulas and limit algebras 5.2 Limit A-Algebras

6 Unification Theorems

1. Introduction

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Quite often relations between sets of elements of a fixed algebraic struc­ture A can be described in terms of equations over A. In the classical case, when A is a field, the area of mathematics where such relations are studied is known under the name of algebraic geometry. It is natural to use the same name in the general case. Algebraic geometry over arbitrary algebraic structures is a new area of research in modern algebra, nevertheless, there are already several breakthrough particular results here, as well as, interest­ing developments of a general theory. Research in this area started with a series of papers by Plotkin [36,37], Baumslag, Kharlampovich, Myasnikov, and Remeslennikov [4,24,25,34].

There are general results which hold in the algebraic geometries over arbitrary algebraic structures, we refer to them as the universal algebraic geometry. The main purpose of this paper is to lay down the basics of the universal algebraic geometry in a coherent form. We emphasize here the relations between model theory, universal algebra, and algebraic geometry. Another goal is quite pragmatic - we intend to unify here some common methods known in different fields under different names. Also, there are several essentially the same results that independently occur in various branches of modern algebra, were they are treated by means specific to the area. Here we give very general proofs of these results based on model theory and universal algebra.

Limit algebras, in all their various incarnations, are the main object of this paper. The original notion came from group theory where limit groups playa prominent part. The limit groups of a fixed group G appear in many different situations: in combinatorial group theory as groups discriminated by G (w-residually G-groups or fully residually G-groups) [2,3,5,6,34]' in the algebraic geometry over groups as the coordinate groups of irreducible varieties over G [4,24-26,47]' groups universally equivalent to G [13,34,40]' limit groups of G in the Grigorchuk-Gromov's metric [10], in the theory of equations in groups [18,24-26,28,38,39]' in group actions [8,12,14,17,35]' in the solutions of Tarski problems [27,48]' etc. These numerous character-

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izations of limit groups make them into a very robust tool linking group theory, topology and logic. It turned out that many of the results on limit groups can be naturally generalized to Lie algebras [19-23].

Our prime objective is to convey some basic facts of the general theory of limit algebras in an arbitrary language. We prove the so-called unification theorems for limit groups that show that the characterization results above hold in the general case as well.

2. Preliminaries

2.1. Languages and structures

Let L = F uP u C be a first-order language (or a signature), consisting of a set F of symbols of operations P (given together with their arities nF), a set P of symbols of predicates P (given together with their arities np) and a set of constants C. If P = 0 then the language L is functional, whereas L is relational if F = C = 0.

For languages L1 ~ L2 we say that L1 is a reduct of L2 and L2 is an expansion of L1' The language L fun = L " P is the functional part of L. From now we fix a first-order functional language L. Almost everything we prove holds (under appropriate adjustments) for arbitrary languages, but the exposition for functional languages is shorter.

Example 2.1. The language of groups consists of a binary operation (multiplication), a unary operation -1 (inversion), and a constant symbol e or 1 (the identity).

Example 2.2. The language of unitary rings consists of three binary oper­ations +, - and· (addition, subtraction and multiplication), and constants o and 1.

An L-structure M is given by the following data: (i) a non-empty set M called the universe of M; (ii) a function pM : Mn F ----t M of arity nF for each F E F; (iii) an element eM E M for each e E C. We often white the structure as M = (M; pM, eM, p E F, e E C). We refer to pM and eM

as interpretations of the symbols P and e in M, and sometimes omit super­scripts M (when the interpretation is obvious from the context). Typically we denote structures in L by capital calligraphic letters and their universes (the underlying sets) by the corresponding capital Latin letters. Structures in a functional language are termed algebras (or universal algebras). An algebra E with the universe consisting of a single element is called trivial. Obviously, interpretation of symbols from L in E is unique.

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As usual, one can define the notion of a homomorphism, and all its variations, between structures in a given language. If a subset N <;: M is closed under the operations F of F and contains all the constants c E C then restrictions of the operations F onto N, together with the constants c, determine a new L:-structure, called a substructure N of M, in which case we write N ~ M. For a subset M' <;: M the intersection of all substructures of M containing M' is a substructure M' of M generated by M' (so M' is a generating set for M'), symbolically M' = (M'). M is termed finitely generated if it has a finite generating set.

Let X = {Xl, X2, .•. } be a finite or countable set of variables. Terms in L: in variables X are formal expressions defined recursively as follows:

Tl) variables Xl, X2, ... , X n , ... are terms; T2) constants from L: are terms; T3) if F(XI, ... ,Xn) E F and tl, ... ,tn are terms then F(tl, ... ,tn ) is

a term.

For F E F we write F(XI, ... , xn) to indicate that n = nF.

By Tc = Tc(X) we denote the set of all terms in L:. For a term t E T.c one can define the set of variables V(t) C X that occur in t. We write t(XI, ... , xn) to indicate that V(t) <;: {Xl, ... , xn}. Also, we use the vector notation t(x), where x = (XI, ... ,Xn). Following the recursive definition of t one can define in a natural way a function tM : Mn ---+ M (which we sometimes again denote by t). If V (t) = 0 then t is a closed term and tM is just a constant. Observe, that the universe of the substructure of M generated by a subset M' <;: M is equal to U{t(M') It E Tc(X)}, where t(M') is the range of the function t.

The condition T3) allows one to define an operation FTc(X) on the set of terms Tc(X). By T2) the set Tc(X) contains all constants from L:, which gives a natural interpretation of constants in T.c(X). These altogether turn the set T .c(X) into an L:-structure T.c(X), which is called the absolutely free L:-algebra with basis X. The name comes from the the following universal property of T.c(X): for any L:-structure M a map h : X ---+ M, extends to a unique L:-homomorphism h : T.c(X) ---+ M.

Formulas in L: (in variables X) are defined recursively as follows:

Fl) if t, sET.c( X) then (t = s) is a formula (called an atomic formula); F2) if ¢ and 'Ij; are formulas then -,¢, (¢ V 'Ij;), (¢ A 'Ij;), (¢ ---+ 'Ij;) are

formulas; F3) If ¢ is a formula and X is a variable then Vx¢ and ~x¢ are formulas.

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For a formula ¢> one can define the set V ( ¢» of free variables of ¢> according to the rules F1)-F3). Namely, V(tl = t2) = V(tl) U V(t2)' V (---,¢» = V(¢», V(¢> 0 'lj;) = V(¢» U V('lj;), where 0 E {V, i\, ----;}, and V(Vx¢» = V(3x¢» = V(¢» ,,{x}. We write ¢>(XI, ... ,Xn) in the case when V(¢» <;;; {XI, ... ,Xn}. Let <I>.c(X) be the set of all formulas in C with V (¢» <;;; X. A formula ¢> with V (¢» = 0 termed a sentence, or a closed formula.

If ¢>(XI, ... ,Xn) E <I>.c(X) and ml, ... ,mn E M then one can define, following the conditions F1)-F3), the relation "¢> is true in M under the in­terpretation Xl ----; ml,.'" Xn ----; m n" (symbolically M F ¢>(ml, ... , m n)). It is convenient sometimes to view this relation as an n-ary predicate ¢>M on M. If h : X ----; M is an interpretation of variables then we denote

¢>h = ¢>M (h(XI)"'" h(xn)). A set of formulas <I> <;;; <I>.c( X) is consistent if there is an C-structure M

and an interpretation h : X ----; M such that M F ¢>h for every ¢> E <I>. In this case one says that <I> is realized in M.

The following result is due to Malcev, it plays a crucial role in model theory.

Theorem [Compactness Theorem] Let K be a class of C-structures and <I> <;;; <I>.c( X). If every finite subset of <I> is realized in some structure in K then the whole set <I> is realized in some ultraproduct of structures from K.

2.2. Theories

Two formulas ¢>, 'lj; E <I> .c(X) are called equivalent if ¢>h = 'lj;h for any inter­pretation h: X ----; M and any C-structure M. One of the principle results in mathematical logic states that any formula ¢> E <I>.c( X) is equivalent to a formula 'lj; in the following form:

(1)

where Qi E {V,3} and 'lj;ij is an atomic formula or its negation. One of the standard ways to characterize complexity of formulas is according to their quantifier prefix QIXI ... Qmxm in (1).

If in (1) all the quantifiers Qi are universal then the formula 'lj; is called universal or V-formula, and if all of them are existential then 'lj; is existential or 3-formula. In this fashion 'lj; is V3-formula if the prefix has only one

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alteration of quantifiers (from V to 3). Similarly, one can define 3V-formulas. Observe, that V- and 3-formulas are dual relative to negation, i.e., the negation of V-formula is equivalent to an 3-formula, and the negation of 3-formula is equivalent to an V-formula. A similar result holds for V3- and 3V­formulas. One may consider formulas with more alterations of quantifiers, but we have no use of them in this paper.

A formula in the form (1) is positive if it does not contain negations (i.e., all'lfJij are atomic). A formula is quantifier-free if it does not contain quantifiers. We denote the set of all quantifier-free formulas from <I> .c(X) by <I>qf,.c(X), and the set of all atomic formulas by At.c(X).

Recall that a theory in the language £ is an arbitrary consistent set of sentences in 12. A theory T is complete if for every sentence ¢ either ¢ or -,¢ lies in T. By Mod(T) we denote the (non-empty) class of all 12-structures M which satisfy all the sentences from T. Structures from Mod(T) are termed models of T and T is a set of axioms for the class Mod(T). Conversely, if K is a class of 12-structures then the set Th(K) of sentences, which are true in all structures from K, is called the elementary theory of K. Similarly, the set Thv(K) (Th3(K)) of all V-sentences (3-sentences) from Th(K) is called the universal (existential) theory of K. The following notions play an important part in this paper. Two 12-structures M and N are elementar­ily equivalent if Th(M) = Th(N) , and they are universally (existentially) equivalent if Thv(M) = Thv(N) (Th3(M) = Th3(N)). In this event we write, correspondingly, M = N, M =v N or M =3 N. Notice, that due to the duality mentioned above M =v N {=} M =3 N for arbitrary 12-structures M and N.

A class of 12-structures K is axiomatizable if K = Mod(T) for some theory Tin 12. In particular, K is V- (3-, or V3-) axiomatizable if the theory T is V- (3-, or V3-) theory.

3. Algebras

There are several types of classes of 12-structures that playa part in general algebraic geometry: prevariaeties, quasivarieties, universal closures, and A­algebras. We refer to [34] for a detailed discussion on this and related mat­ters. Here we present only a few properties and characterizations of these classes, that will be used in the sequel. Most of them are known and can be found in the classical books on universal algebra, for example, in [30]. On the algebraic theory of quasivarieties, the main subject of this section, we refer to [15].

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3.1. Congruences

In this section we remind some notions and introduce notation on presen­tation of algebras via generators and relations.

Let M be an arbitrary fixed C-structure. An equivalence relation () on M is a congruence on M if for every operation P E F and any elements m1, ... ,mnF , m~, ... ,m~F EM such that mi rvB m~, i = 1, ... ,np, one has pM(m1, .. " m nF ) rvB pM(m~, ... , m~F)'

For a congruence () the operations pM, P E F, naturally induce well­defined operations on the factor-set M/(). Namely, if we denote by m/() the equivalence class of mE M then pMjB is defined by

pMjB(ml/(), ... , m nF /()) = pM(m1, ... , m nF )/()

for any m1,"" m nF E M. Similarly, cM / B is defined for c E C as the class cM /(). This turns the factor-set M/() into an C-structure. It follows immediately from the construction that the map h : M ---4 M / (), such that h(m) = m/(), is an C-epimorphism h : M ---4 M/(), called the canonical epimorphism.

The set Con(M) of all congruences on M forms a lattice relative to the inclusion ()1 ~ ()2, i.e., every two congruences in Con(M) have the least upper and the greatest lower bounds in the ordered set (Con(M), ~). To see this, observe first that the intersection of an arbitrary set e = {()i, i E I} of congruences on M is again a congruence on M, hence the greatest lower bound for e. Now, the intersection of the non-empty set {() E Con(M) I ()i ~ () 'V ()i E e} is the least upper bound for e. The following result is easy.

Lemma 3.1. Let M be an C-algebra, {()i liE I} ~ Con(M) and

() = niEI Bi· Then M/B embeds into the direct product DiE I M/()i via the diagonal monomorphism m/B ---4 DiE I m/()i.

A homomorphism h : M ---4 N of two C-structures determines the kernel congruence ker h on M, which is defined by

m1 rvkerh m2 {==} h(m1) = h(m2), m1, m2 EM.

Observe, that if () E Con(M) and B ~ ker h then the map h : M/() ---4 N defined by h(m/()) = h(m) for m E M is a homomorphism of C-structures.

Definition 3.1. A set of atomic formulas t:. ~ Atc(X) is called congruent if the binary relation ()t::. on the set of terms Tc(X) defined by (where t1,t2 E T.c(X))

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is a congruence on the free .c-algebra T.c(X).

The following lemma characterizes congruent sets of formulas.

Lemma 3.2. A set of atomic formulas ~ ~ At.c(X) is congruent if and only if it satisfies the following conditions:

(1) (t = t) E ~ for any term t E T.c(X); (2) if (tl = t2) E ~ then (t2 = tI) E ~ for any terms h, t2 E Tc(X); (3) if (h = t2) E ~ and (t2 = t3) E ~ then (h = t3) E ~ for any terms

tl, t2, t3 E T.c(X); (4) if (h = SI), ... ,(tnF = snF) E ~ then (F(tl, ... ,tnF )

F(SI, ... ,SnF)) E ~ for any terms ti,Si E T.c(X), i = 1, ... ,nF, and any functional symbol F E .c.

Proof. Straightforward. o

Since the intersection of an arbitrary set of congruent sets of atomic formulas is again congruent, it follows that for a set ~ ~ Atc(X) there is the least congruent subset [~] ~ Atc(X), containing ~. Therefore, ~ uniquely determines the congruence B t::. = B[t::.].

For an .c-algebra M generated by a set M' ~ M put X = {xm I m E

M'} and consider a set ~M' of all atomic formulas (tl = t2) E At.c(X) such that M F (h = t2) under the interpretation Xm -+ m, m EM'. Obviously, ~M' is a congruent set in At.c(X) (the set of all relation in M relative to M'). A subset 5 ~ ~M' is called a set of defining relations of M relative to M' if [5] = ~M" In this event the pair (X I 5) termed a presentation of M by generators X and relations 5.

Lemma 3.3. If (X I 5) is a presentation of M then M ~ Tc(X)/Bs.

Proof. The map h' : X -+ M' defined by h'(xm ) = m, m EM', extends to a homomorphism h : T.c(X) -+ M. Clearly, tl "'kerh t2 if and only if (tl =

t2 ) E [5] for terms t l , t2 E T.c(X). Therefore, T.c(X)/Bs ~ T.c(X)/ ker h. Now the result follows from the isomorphism T.c(X)/ ker h ~ M. 0

3.2. Quasivarieties

In this section we discuss quasivarieties and related objects. The main focus is on how to generate the least quasi variety containing a given class of structures K. A model example here is the celebrated Birkhoff's theorem which describes Var(K), the smallest variety containing K, as the class

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HSP(K) obtained from K by taking direct products (the operator P), then substructures (the operator S), and then homomorphic images (the operator H). Along the way we introduce some other relevant operators. On the algebraic theory of quasivarieties we refer to [15] and [30].

We fix, as before, a functional language £ and a class of £-algebras K. We always assume that K is an abstract class, i.e., with any algebra M E K the class K contains all isomorphic copies of M.

Recall that an identity in £ is a formula of the type

where t, s are terms in £. Meanwhile, a quasi-identity is a formula of the type

where t(x), sex), ti(X), Si(X) are terms in £ in variables x = (Xl, ... , xn). A class of £-structures is called a quasivariety (variety) if it can be

axiomatized by a set of quasi-identities (identities). Given a class of £­structures K one can define the quasivariety Qvar(K), generated by K, as the quasivariety axiomatized by the set Thqj(K) of all quasi-identities which are true in all structures from K, i.e., Qvar(K) = Mod(Thqj(K)). Notice, that Qvar(K) is the least quasivariety containing K. Similarly, one defines the variety Var(K) generated by K.

Observe, that an identity Vx(t(x) = sex)) is equivalent to a quasi­identity V x( X = X --t t( x) = s( x)), therefore, Qvar(K) ~ Var(K).

Before we proceed with quasivarieties, we introduce one more class of structures. Namely, K termed a prevariety if K = SP(K). By Pvar(K) we denote the least prevariety, containing K. The prevariety Pvar(K) grasps the residual properties of the structures from K. An £-structure M is separated by K if for any pair of non-equal elements ml, m2 E M there is a structure N E K and a homomorphism h : M --t N such that h( ml) "I=­

h(m2)' By Res(K) we denote the class of £-structures separated by K. In the following lemma we collect some known facts on prevarieties.

Lemma 3.4. For any class of £-structures K the following holds:

1) Pvar(K) = SP(K) ~ Qvar(K); 2) Pvar(K) = Res(K); 3) Pvar(K) is axiomatizable if and only if Pvar(K) = Qvar(K).

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Proof. Equality 1) follows directly from definitions. 2) was proven for groups in [34], here we give a general argument. It

is easy to see that Res(K) is a prevariety, so Pvar(K) ~ Res(K). To show converse, take a structure M E Res(K) and consider the set I of all pairs (ml,m2), ml,m2 E M, such that ml =I- m2· Then for every i E I there exists a structure M E K and a homomorphism hi : M ----) M with hi(ml) =I- hi(m2). The homomorphisms hi, i E I, give rise to the "diagonal" homomorphism h : M ----) Ilo M, which is injective by construction. Hence M E SP(K), as required.

3) is due to Malcev [31]. 0

Prevarieties play an important role in combinatorial algebra, they can be characterized as classes of structures admitting presentations by generators and relator. Namely, let X be a set and ~ a set of atomic formulas from <I>c(X). Following Malcev [30], we say that a presentation (X I ~) defines a structure M in a class K if there is a map h : X ----) M such that

D1) h(X) generates M and all the formulas from ~ are realized in M under the interpretation h;

D2) for any structure N E K and any map f : X ----) N if all the formulas from ~ are realized in N under f then there exists a unique homomor­phism g : M ----) N such that g(h(x)) = f(x) for every x E X.

If (X I ~) defines a structure in K then this structure is unique up to isomorphism, we denote it by FK(X, ~).

Theorem [30] A class K, containing the trivial system £, is a prevariety if and only if any presentation (X I ~) defines a structure in K.

To present similar characterizations for quasivarieties we need to intro­duce the following operators.

As was mentioned above, P(K) is the class of direct products of struc­tures from K. Recall, that the direct product of C-structures M i , i E I, is an C-structure M = DiE! Mi with the universe M = DiE! Mi where the functions and constants from C are interpreted coordinate-wise. If all the structures Mi are isomorphic to some structure N then we refer to DiE! Mi as to a direct power of N and denote it by N!, By Pw(K) we denote the class of all finite direct products of structures from K.

Recall, that a substructure N of a direct product DiE! Mi is a subdirect product of the structures M i, i E I, if pj(N) = M j for the canonical

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projections Pj : I1iEI Mi -- Mj , j E I. By P,.(K) we denote the class of all subdirect products of structures from K.

Let I be a set, D a filter over I (i.e., a collection D of subsets of I closed under finite intersections and such that if a E D then bED for any b <::; I with a <::; b, and also we assume that 0 rf. D), and {Mi liE I} a family of sets. On the direct product I1iEI Mi one can define an equivalence relation "'"'D such that a "'"'D b if and only if {i E I I pi(a) = Pi(b)} ED. We denote the factor-set by I1iEI Mil D, and the equivalence class of an element a by aiD. Now, if {Mi liE I} is a collection of £-structures then the equivalence"'"' D becomes a congruence on the direct product I1iEI M i, in which case the filterproduct M = I1iEI MilD of the structures M i, i E I, over D is defined as the factor-structure I1iEI Mil'" D. If D is an ultrafilter on I (a filter that contain either a or I" a for any a <::; I) then a filterprod uct over D is called an ultraproduct, furthermore, if all the structures Mi are isomorphic to some structure N then the ultraproduct I1iEI MilD is called an ultrapower and we denote it by NI I D. By PdK) and Pu(K) we denote, correspondingly, the classes of filterproducts and ultraproducts of structures from K.

Let Ke = K U {E}, where E is the trivial £-structure introduced earlier. A word of warning is needed here. Sometimes, direct products I1iEI Mi are defined being equal to E for the empty set I (see, for example, [15]), but we elect not to do so, assuming always that I is non-empty and adding [, to the class, if needed.

Lemma 3.5. For any class of £-structures K the following holds:

5) Qvar(K) = SPr(K) e; 6) Qvar(K) = SPPu(K) e = SPuP(K) e;

7) Qvar(K) = SPuPw(K) e;

Proof. 5) is due to Malcev [30]([§11, Theorem 4]). 6) and 7) are due to Gorbunov [15]([Corollary 2.3.4, Theorem 2.3.6]). 0

Now we give another characterization of quasivarieties, for this we need to introduce direct limits.

Recall, that a partial ordering (1,~) is directed if any two elements from I have an upper bound. A triple A = (I,Mi,h ij ), consisting of a directed ordering (I, ~), a set of £-structures {M i , i E I}, and a set of homomorphisms hij : Mi -- M j (i,j E I, i ~ j), is called a direct system of structures M i , i E I, if

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(1) h ii is the identity map for every i E I;

(2) hjk 0 hij = hik for any i, j, k E I with i ~ j ~ k.

We call a directed system A = (1, M i , hij ) epimorphic if all the homomor­phisms hij : Mi -> M j are surjective.

Given a direct system A = (1, M i , hij ) one can consider an equivalence relation == on a set {(mi' i) I mi E M i , i E I} defined by

(mi,i) == (mj,j) {:} :J k E I, i,j ~ k, hik(mi) = hjk(mj).

By (m, i) we denote the equivalence class of (m, i) under ==. Now one can turn the factor-set M = {( mi, i) I mi E M i , i E I} / == into an .c-structure M interpreting the constants and functions from .c as follows:

(1) if e E.c is a constant then eM = (eMi,i) for an arbitrary chosen i E I; (2) if F E .c is a function and (ml' il), ... , (mnF' inF ) E M then

FM ((ml' il), ... , (mnF , inF )) = FMj ((hid(ml), i l ), ... , (hinFj(mnF)' inF ))

for an arbitrary chosen j E I with iI, ... ,inF ~ j.

The structure M is well-defined, it is called the direct limit of the system A, we denote it by lli!}Mi' It is easy to see that lli!}Mi has the following property. Let i E I be a fixed index. Put Ji = {j E I I i ~ j} a nd denote Ai = (h M j , hjk' j, k E Ji ). Then Ai is a direct system whose direct limit Mi is isomorphic to M. By L (K) and Ls(K) we denote the class of direct

-> -> and epimorphic direct limits of structures from K.

The following result gives a characterization of quasivarieties in terms of direct limits.

Lemma 3.6. For any class of .c-structures K the following holds:

Proof. See [15] ([Corollary 2.3.4]). o

3.3. Universal closures

In this section we study the universal closure Ucl(K) = Mod(Thv(K)) of a given class of .c-structures K.

Structures from Ucl(K) are determined by local properties of structures from K. To explain precisely we need to introduce two more operators.

Recall [5,34]' that a structure M is discriminated by K if for any finite set W of elements from M there is a structure N E K and a homomorphism

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h : M ---., N whose restriction onto W is injective. Let Dis(K) be the class of .c-structures discriminated by K. Clearly, Dis(K) ~ Res(K).

To introduce the second operator we need to describe local submodels of a structure M. First, we replace the language .c by a new relational language .cre!, where every operational and constant symbols F E :F and c E C are replaced, correspondingly, by a new predicate symbol RF of arity nF + 1 and a new unary predicate symbol Re. Secondly, the structure M

l l . MTel MTel turns into a .cre -structure Mre , where the predIcates Re and RF

are defined by

R1) for m E M the predicate R.;tTel (m) is true in Mrel if and only if eM =m;

R2) for mo, ml,"" m nF E M the predicate R-j;:lTel (mo, ml,"" m nF ) is true in Mrel if and only if FM(ml,'" ,mnF ) = mo.

Third, if .co is a finite reduct (sublanguage) of .c then by Meo we de­note the reduct of Mrel, where only predicates corresponding to constants and operations from .co are survived, so Meo is an .cae/-structure. Now, following [30], by a local submodel of M we understand a finite substructure of Meo for some finite reduct .co of .c.

Finally, a structure M is locally embeddable into K if every local sub­model of M is isomorphic to some local submodel of a structure from K (in the language .coel ). By L(K) we denote the class of .c-structures locally embeddable into K.

It is convenient for us to rephrase the notion of a local submodel in terms of formulas.

Let .c' be a finite reduct of.c and X a finite set of variables. A quantifier­free formula cp in .c' is called a diagram-formula if cp is a conjunction of atomic formulas or their negations that satisfies the following conditions:

1) every formula -,(x = y), for each pair (x,y) E X 2 with x i= y, occurs in cp;

2) for each functional symbol F E .c' and each tuple of variables (XO,Xl, ... ,XnF ) E xnF+I either formula F(XI,".,XnF ) = Xo or its negation occurs in cp;

3) for each constant symbol c E .c' and each x E X either x = c or its negation -,(x = c) occurs in cpo

We say that cp is a diagram-formula in .c if it is a diagram-formula for some finite reduct .c' of.c and a finite set X. The name of diagram-formulas comes from the diagrams of algebraic structures (see Section 3.4).

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The following lemma is easy.

Lemma 3.7. For any local submodel N of M there is a diagram-formula 'PN(X) in a finite set of variables X of cardinality [N[ such that M F 'PN(h(X)) for some bijection h : X ---4 N. And conversely, if M F 'P(h(X)) for some diagram-formula 'P(X) in C and an interpretation h : X ---4 M then there is a local submodel N of M with the universe heX) such that 'P = 'PN (up to a permutation of conjuncts).

Corollary 3.1. An .c-structure M is locally embeddable into a class K if and only if every diagram-formula realizable in M is realizable also in some structure from K.

Lemma 3.8. For any class of .c-structures K the following holds:

8 Ucl(K) = L(K); 9 U cl(K) = SPu (K);

10 Dis(K) <;;;; Ucl(K); 11 l(K) <;;;; Ucl(K).

Proof. To prove 8) and 9) we show that L(K) <;;;; SPu(K) <;;;; Ucl(K) C

L(K). The first inclusion has been proven by Malcev [30], but we briefly discuss it for the sake of completeness. Let M be a structure from L(K). By Corollary 3.1 every diagram-formula 'P realizable in M is realizable also in some structure Ncp from K. By the Compactness Theorem the set <PM of all diagram-formulas realizable in M is realized in some ultraproduct N = I1cp Ncp/ D, where 'P runs over <PM. By Lemma 3.9 the core Diago(M) of the diagram of M is also realized in N under an appropriate interpretation of constants cm , m E M (see Section 3.4). Now the substructure of N generated by all elements em, m EM, is isomorphic to M. Hence M E

SPu(K). Inclusion SPu(K) <;;;; Ucl(K) follows from two known results: any uni­

versal class is closed under substructures (which is obvious) and the Los theorem [30,32]. To see that Ucl(K) <;;;; L(K) consider an arbitrary M E Ucl(M). If 'P(Xi, ... , xn) is a diagram-formula which is realized in M then a universal sentence 'I/; = 'lixi, ... ,xn ....., 'P(Xi, ... ,xn ) is false in M. Hence, there exists a structure N E K on which 'I/; is false, so N F .....,'1/;. Therefore, 'P(Xi, ... , xn) is realized in N. By Corollary 3.1 M E L(K), as required.

To see 10) it suffices to notice that Dis(K) <;;;; L(K) and then apply 8).

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11) follows from 9) and [15J (Theorem 1.2.9), where it is shown that

.1(K) ~ SPu(K). 0

3.4. A-Algebras

Let A be a fixed C-algebra. In this section we discuss A-algebras - principal objects in algebraic geometry over A. Informally, an A-algebra is an C­algebra with a distinguished sub algebra A. Even though this notion seems simple, one needs to develop a formal framework to deal with A-algebras. It will be convenient to use two equivalent approaches: one is categorical and another is logical (or axiomatic).

Definition 3.2. [CategoricalJ An A-algebra is a pair (B, A), where B is an C-algebra and A : A --. B is an embedding.

For the axiomatic definition we are going to use the language of dia­grams. By CA we denote the language C U {ca I a E A}, which is obtained from C by adding a new constant Ca for every element a E A.

Observe, that every A-algebra (B, A) can be viewed as an CA-algebra when the constant Ca is interpreted by A(a).

Recall that by At.cA (0) we denote the set of all atomic sentences in the language CA. The diagram Diag(A) of A is the set of all atomic sentences from At.cA (0) or their negations which are true in A. To work with diagrams we need to define several related sets of formulas.

The core Diago(A) of the diagram Diag(A) consists of the following formulas:

• C = Ca for each constant symbol C E C and a E A such that cA = a; • P( Ca" ... 'canF ) = cao ' for each functional symbol P E F and each tu­

ple of elements (ao, aI, ... ,anF ) E AnF+I such that pA(al' ... ,anF ) = ao;

• Ca, i=- Ca2 , for each pair (aI, a2) E A2 such that al i=- a2.

The following result is easy

Lemma 3.9. For an C-algebra A the following hold:

Cl) For every CA-structure B if B F Diago(A) then B F Diag(A),-C2) If S is a finite subset of Diago (A) then there is a diagram-formula 'P(X)

in C and an interpretation h : X --. A such that every formula from S occurs as a conjunct in 'P(h(X)) (after replacing h(x) with Ch(x)) and A F 'P(h(X)),-

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C3) If cp(X) is a diagram-formula in Land h : X -+ A is an interp'1:fta­tion such that A 1= cp(h(X)) then every conjunct of cp(X) (where x is replaced with Ch(x)) belongs to Diag(A).

The following result gives an axiomatic way to describe A-algebras.

Lemma 3.10. Let B be an L-algebra and .\ : A -+ B a map. Then (B,.\) is an A-algebra if and only if B 1= Diag(A), where Ca is interpreted by .\(a) for every a E A.

Proof. Straightforward. o

This leads to the following, equivalent, definition of A-algebras.

Definition 3.3. [Axiomatic] An algebra B in the language LA is called an A-algebra if B 1= Diag(A).

Put

Let Cat(A) be the class of all A-algebras. Since A-algebras are LA­structures the standard notions of a LA-homomorphism, LA-substructure, LA-generating set, etc., are defined in Cat(A). Sometimes, we refer to them as to an A-homomorphism, A-substructure, A-generating set, etc.

All the operators 0 introduced in Sections 3.3 and 3.2 are defined for LA-structures, but, a priori, the resulting LA-algebra may not be in the class Cat(A). Nevertheless, one can check directly for each such operator 0 (with the exception of the operator K -+ Ke that adds the trivial structure [. to K) that O(Cat(A)) ~ Cat(A). Sometimes, we add the subscript A and write 0 A to emphasize the fact that the algebras under consideration are A-algebras. Another, shorter, way to prove this is to show that Cat(A)e is a quasivariety, and then these results, as well as some others, will follow for free.

Lemma 3.11. The class Cat(A) e is a quasivariety in the language LA defined by the following set of quasi-identities:

(1) C = Ca , for each constant symbol C ELand element a E A such that a = cA ,.

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(2) F(cal , ... , canF ) = ca, for each functional symbol FE L and each tuple

(al, ... ,anF,a) E AnF+I such that FA(al' ... ,anF ) = a;

(3) \:j x \:j Y (Cal = Ca2 ----) X = y), for each pair of elements aI, a2 E A with

UI =I U2·

Proof. It is easy to see that any A-algebra and the trivial algebra E sat­isfy the formulas above. One needs to check the converse. Suppose C is an LA-algebra, satisfying the formulas above. If C = E then C E Cat(A) e. As­sume now that C =I E. The formulas 1) and 2) show that C F Diago(A) n Diag+ (A), while the formulas 3) provide C F Diago(A) n Diag- (A). Alto­gether, C F Diago(A), so by Lemma 3.9 C F Diag(A), as claimed. 0

Corollary 3.2. Let A be an algebra and K a class of A-algebras. Then the

following holds:

1) K is closed under the operators SA, P A, PwA , P..A, PfA, PuA , .hA

,

.hs A' LA; 2) every algebra in the classes PvarA(K), UclA(K), ResA(K), and

DisA(K) is an A-algebras;

3) every algebra in QvarA(K), with the exception of E, is an A-algebra.

4. Types, Zariski topology, and coordinate algebras

In this section we introduce algebras defined by complete atomic types.

4.1. Quantifier-free types and Zariski topology

Let L be a functional language, T a theory in L, and X = {Xl, ... ,Xn } a finite set of variables. Recall (see, for example, [32]), that a type in variables X of Lover T is a consistent with T set p offormulas in cI> c(X), i.e, a subset p ~ cI>c(X) that can be realized in a structure from Mod(T).

A type p is complete if it is a maximal type in cI> c(X) with respect to inclusion. It is easy to see that if p is a maximal type in X then for every formula i.p E cI> c( X) either i.p E P or --, i.p E p.

Definition 4.1. A set p of atomic or negations of atomic formulas from cI> c(X) is called an atomic type in X relative to a theory T if pUT is consistent. A maximal atomic type in cI> c( X) with respect to inclusion termed a complete atomic type of T.

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It is not hard to see that if p is a complete atomic type then for every atomic formula 'P E At.c(X) either 'P E P or -, 'P E p.

Example 4.1. Let M be an .c-structure and m = (ml,"" m n ) E Mn. Then the set atpM(m) of atomic or negations of atomic formulas from <I>.c (X) that are true in M under an interpretation Xi f---+ mi, i = 1, ... , n,

is a complete atomic type relative to any theory T such that M E Mod(T).

We say that a complete atomic type p in variables X is realized in M if p = atpM(m) for some m E Mn.

Every type p in T can be realized in some model of T (i.e., a structure from Mod(T)). If p cannot be realized in a structure M then we say that M omits p. There are deep results in model theory on how to construct models of T omitting a given type or a set of types.

For an atomic type p ~ <I> .c(X) by p+ and p- we denote, correspond­ingly, the set of all atomic and negations of atomic formulas in p.

If S is a set of atomic formulas from <I> .c(X) and M is an .c­structure then by VM(S) we denote the set {(ml, ... ,mn ) E Mn 1M 1= S(ml, .. " mn )} of all tuples in Mn that satisfy all the formulas from S. The set V M (S) is called the algebraic set defined by S in M. We refer to S as a system of equations in .c, and to elements of S - as equations in .c. Sometimes, to emphasize that formulas are from .c we call such equations (and systems of equations) coefficient-free equations, meanwhile, in the case when .c = .cA, we refer to such equations as equations with coefficients in algebra A.

Following [4] we define Zariski topology on M n , n ~ 1, where algebraic sets form a prebasis of closed sets, i.e., closed sets in this topology are ob­tained from the algebraic sets by finite unions and (arbitrary) intersections.

If p is an atomic type in .c in variables X = {Xl, ... , xn} then V M (p+)

is an algebraic set in Mn. More generally, for an arbitrary type p in X by p+ we denote the set of all positive formulas in p, i.e., all formulas in the prenex form that do not have the negation symbol.

If p is quantifier-free type, i.e., a type consisting of quantifier-free for­mulas, then formulas in p+ are conjunctions and disjunctions of atomic formulas.

Lemma 4.1. Let M be an .c-structure and n E N. Then for a subset

V ~ Mn the following conditions are equivalent:

• V is closed in the Zariski topology on Mn,-

• V = V M(P+) for some quantifier-free type p in variables {Xl, ... , x n }.

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Proof. Straightforward. o

4.2. Coordinate algebras and complete types

Let M be an .c-algebra. For a set 8 of atomic formulas from q,dX) denote by RadM (8) the set of all atomic formulas from q,dX) that hold on every tuple from V M(8). In particular, if V M(8) = 0 then RadM (8) = Atc(X). It is not hard to see that RadM (8) is a congruent set of formulas, hence it defines a congruence that we denote by BRad(S). The .c-structure Tc(X)jBRad(S) is called the coordinate algebra of the algebraic set V M (8). If Y = V M(8) then the coordinate algebra TdX)jBRad(S) is denoted by feY) and Rad(8) - by Rad(Y).

The following result gives a characterization of the coordinate algebras over an algebra M.

Proposition 4.1. A finitely generated .c-algebra C is the coordinate algebra of some non-empty algebraic set over an .c-algebra M if and only if C is separated by M.

Proof. Let Y be an algebraic set in Mn. With a point p = (ml' ... ,mn) E

M n we associate a homomorphism hp : Tc(X) ----; M defined by hp(t) = tM(ml, ... , m n ). Clearly,

BRad(Y) = n ker hp­pEY

Therefore, the diagonal homomorphism I1pEY : Tc(X) ----; I1PE

Y M induces a monomorphism

f(Y) = TdX)jBRad(Y) ----; MIYI.

It follows that r(Y) E SP(M). Now, by Lemma 3.4 SP(M) = Res(M), so feY) E Res(M).

Suppose now that C is a finitely generated .c-algebra from Res(M) with a finite generating set X = {Xl, ... , x n }. Let C = (X I 8) be a presentation of C by the generators X and relations 8 ~ Atc(X). In this case C is isomorphic to Tc(X)jBs. To prove that C is the co­ordinate algebra of some algebraic set over M it suffices to show that RadM (8) = [8]. If (tl = t2) (j. [S] then there exists a homomorphism h : C ----; M with tj"1(h(XI), ... , h(xn)) i= t~(h(XI)' ... ' h(xn)). Obviously,

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(h(XI),"" h(xn)) E V M(S) so (tl = t2) rf- RadM(S). This shows that RadM(S) = [S]. D

Lemma 4.2. Let p be a complete atomic type in variables X. Then:

• p+ is a congruent set of formulas; • p+ = RadM(p+) for every £-structure M with V M(P) =1= 0.

Proof. Indeed, since p is realized in some model M of T its positive part p+ satisfies the assumptions of Lemma 3.2, hence it is congruent. It follows that p+ determines a congruence ()p on Tc(X). Since p is complete one has p+ = RadM (p+). D

Definition 4.2. Let X be a finite set of variables and p a complete atomic type in variables X. Then the factor-algebra Tc(X)/()p of the free £­algebra Tc(X) is termed the algebra defined by the type p and the tuple (xI/()p, ... ,Xn/()p) is called a generic point of p.

Clearly, any complete atomic type p in variables X in a theory T is realized in the factor-algebra Tt:,(X)/()p at the generic point x (XI/()p,""xn/()p), so

atpTcCX)/(lp(X) = p.

Indeed, for any atomic formula tl = t2, where tl, t2 E T,dX) one has (h = t2) E p if and only if h "'(lp t2, which is equivalent to the condition Tc(X)/()p F (h = t2) under the interpretation Xi I----' xi/()p' The generic point (xI/()p,"" xn/()p) satisfies the following universal property. If p is realized in some £-structure Mat (ml,"" mn) E Mn then the map Xl -. ml,"" Xn -. mn extends to a homomorphism T.c(X)/()p -. M.

Lemma 4.3. Let T be a universally axiomatized theory in.c. Then for any finitely generated £-structure M the following conditions are equivalent:

1) M E Mod(T); 2) M = T.c(X)/()p for some complete atomic type p in T.

Proof. Let X = {Xl, ... , xn} be a finite set and (X I S) a presentation of an £-structure M, i.e., M ~ Tc(X)/()s. If p = atpM(x), X = (Xl, ... , xn) then [S] = p+ and Tt:..(X)/()p ~ Tc(X)/()s ~ M. Therefore, 1) implies 2).

To prove the converse, let p be an atomic type in T. We need to show that Tc(X)/()p E Mod(T). Since p is a type in T there exists a model N E Mod(T) and a tuple of elements fj = (YI,"" Yn) E Nn such that

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p = atpN(y). If N' is a substructure of N generated by Yl,'" ,Yn then T.c(X)jBp ~ N'. Since the theory T is axiomatized by a set of universal sentences one has N' E Mod(T). Hence, Tt:,(X)jBp E Mod(T). 0

4.3. Equationally Noetherian algebras

The notion of equationally Noetherian groups was introduced in [4] and [7]. Let B be an algebra. For every natural number n we consider Zariski

topology on Bn. A subset Y <;;:; Bn is called reducible if it is a union of two proper closed

subsets, otherwise, it is called irreducible. It is not hard to see that an algebraic set Y <;;:; Bn is irreducible if and

only if it is not a finite union of proper algebraic subsets. Recall, that a topological space is called Noetherian if it satisfies the

descending chain condition on closed subsets.

Remark 4.1. Let (W, 'r) be a topological space, m a prebase of closed subsets of '!, and SJ3 the base of T, formed by the finite unions of sets from m. Suppose that m is closed under finite intersections. Then the following conditions are equivalent:

• the topological space (W, '!) is Noetherian; • m satisfies the descending chain condition.

In this case

1) the base SJ3 contains all closed sets in the topology 'I; 2) any closed set Y in '! is a finite union of irreducible closed sets from

m (irreducible components): Y = Y1 U ... U Y m' Moreover, if Yi ~ lj for i #- j then this decomposition is unique up to a permutation of components.

Definition 4.3 (No coefficients). An algebra B is equationally Noethe­rian, if for any natural number n and any system of equations 8 <;;:;

At.c(Xl, ... , xn) there exists a finite subsystem 80 <;;:; 8 such that V 8(8) = V8 (80).

Definition 4.4 (Coefficients in A). An A-algebra B is A-equationally Noetherian if for any natural number n and any system of equations 8 <;;:; AtcA (Xl, ... , xn) there exists a finite subsystem 80 <;;:; 8 such that V 8(8) = V 8(80 ),

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Lemma 4.4. An (A-) algebra B is (A-) equationally Noetherian if and only if for any natural number n Zariski topology on B n is Noetherian.

Proof. We prove the lemma for coefficient-free equations, a similar argu­ment gives the result for equations with coefficients in A.

Assume B is equationally Noetherian and consider a descending chain of closed subsets Y1 2 Y2 2 Y3 2 ... of algebraic sets in Bn. Taking the radicals one gets an ascending chain of subalgebras Rad(Y1) ~ Rad(Y2 ) ~

Rad(l3) ~ .... Put S = U Rad(Yi). By our assumption the system S is equivalent to some finite subsystem So ~ S. Clearly, So ~ Rad(Yi) for some index i. Therefore, the chains before stabilize.

Suppose now that for any natural number n Zariski topology on B n is Noetherian. Let S ~ At.c(X1, ... , xn) be an arbitrary system of equations in variables {X1, ... ,Xn }. Let (t1 = 81) E S. IfV6(8) = V6({h = sd) then there is nothing to prove. Otherwise, there is an atomic formula (t2 = 82) E 8\{h = sd with V({t1 = sd) ;2 V({t1 = 81,t2 = sd). Repeating this process one can produce a descending chain of closed subsets in Bn. Since Bn is Noetherian the chain is finite, so V 6(8) = V 6(80 ) for some finite subsystem 80 of 8. 0

The following result follows immediately from Lemma 4.4 and Re­mark 4.1.

Theorem 4.1. Let B be an (A-)equationally Noetherian (A-)algebra. Then any algebraic set Y ~ B n is a finite union of irreducible algebraic sets (irreducible components),' Y = Y1 U ... U Y m . Moreover, if Yi ~ Yj for i -=I- j then this decomposition is unique up to a permutation of components.

Now we give a characterization of the coordinate algebras of irreducible algebraic sets over an arbitrary algebra B.

Lemma 4.5. Let Y be an irreducible algebraic set over B. Then the coor­

dinate algebra f(Y) is discriminated by B.

Proof. Indeed, let Y = V(S) and f(Y) = T.c(X)/BRad(Y). Suppose, to the contrary, that there exist such atomic formulas (ti = 8i) E At.c(X), (ti = 8i) rf- Rad(Y), i = 1, ... , m, such that for any homomorphism h : r(Y) --t B there exists an index i E {I, ... , m} for which h(ti/BRad(s) = h(sdBRad(S)' This implies that for any p E Y there exists an index i E {I, ... , m} with t~ (p) = 8~ (p). Put Yi = V (8 U {ti = 8d), i = 1, ... , m. Then Y = Y1 U ... U

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Ym and the sets Y1 , ... , Ym are proper closed subsets of Y - contradiction with irreducibility of Y. This shows that f(Y) is discriminated by B. 0

The converse of this result also holds.

Lemma 4.6. Let C be a finitely generated .c-algebra. If C is discriminated by an .c-algebra B then C is the coordinate algebra of some algebraic set

over B.

Proof. Since Dis(B) <;;; Res(B) then by Proposition 4.1 C = f(Y) for some algebraic set Y over B. To prove the result it suffices to reverse the argument in Lemma 4.5. Indeed, suppose Y = Y1 U ... U Y m for some proper algebraic subsets Yi. From Yi C Y and Yi i= Y follows that Rad(Y) C

Rad(Yi) and Rad(Y) i= Rad(Yi), so there exists an atomic formula (ti =

Si) E Rad(Yi)\Rad(Y), i = 1, ... , m. This implies that there is no any

homomorphism h : r(Y) ~ B with h(ti/BRad(Y») i= h(si/BRad(Y») for all i = 1, ... , m, - contradiction with C E Dis(B). 0

Theorem 4.2. Let B be an .c-algebra and C a finitely generated .c-algebra.

Then C is the coordinate algebra of some irreducible algebraic set over B if and only if C is discriminated by B.

Proof. Follows immediately from Lemmas 4.5 and 4.6, and Remark 4.1.0

A similar argument gives the result for A-algebras.

Theorem 4.3. Let B be an A-algebra and C a finitely generated A-algebra.

Then C is the coordinate algebra of some irreducible algebraic set over B if and only if C is A-discriminated by B.

5. Limit algebras

5.1. Direct systems of formulas and limit algebras

In this section we discuss limit .c-algebras. We need the following notation. For a formula rp E <I> c(X) and a map 'Y : X ~ XI from X into a set of variables XI by rp("((X)) we denote the formula obtained from r.p by the substitution x ~ 'Y(x) for every x E X.

Definition 5.1. A triple A = (I, rpi, 'Yij) is called a direct system of for­

m ulas in .c if

(1) (J,~) is a directed ordering;

103

(2) for each i E I there is a finite reduct .ci of.c and a finite set of variables Xi such that 'Pi is a consistent diagram-formula in .ci in variables Xi;

(3) "tij : Xi ----+ X.i is a map defined for every pair of indices i, j E I, i ~ j, such that:

• "tii is the identical map for every i E I; • "tjk 0 "tij = "tik for every i,j, k E I, i ~ j ~ k; • all conjuncts of 'Pi("(ij(Xi )) are also conjuncts of 'Pj(Xj );

(4) for any c E .c there exists i E I such that 'Pi contains a conjunct of the type Xi = c, where Xi E Xi;

(5) for any functional symbol P E .c, any i E I, and any tuple of variables (Xl, ... ,XnF ) E X;F there is j E I, i ~ j such that 'Pj contains a conjunct of the type P("(ij(Xl), ... , "tij (xnF )) = Xj, where Xj E Xj'

Let A = (I, 'Pi, "tij) be a direct system of formulas in .c. Define a factor-set L(A) = {(Xi, i), Xi E Xi, i E I}/ ==, where

(xi,i) == (Xj,j) {? 3 k E I, i,j ~ k, "tik(Xi) = "tjk(Xj).

By (x, i) we denote the equivalence class of an element (x, i), X E Xi, i E I, relative to ==.

We turn the set L(A) into an .c-algebra interpreting constants and op­erations from .c on L(A) as follows:

(1) if c E .c is a constant symbol then cL(A) = (Xi, i), where i E I is an arbitrary index such that the conjunction 'Pi contains an atomic formula of the type Xi = c, with Xi E Xi;

(2) if P E .c is a symbol of operation and (xl,il)"",(XnF,inF) E

L(A) then pL(A)(Xl,il)"",(XnF,inF)) = (Xj,j), where j E I, i l , ... , inF ~ j and such that the conjunction 'Pj contains a conjunct P("(id(Xl), ... , "tinFj(xnF)) = Xj, Xj E Xj'

Lemma 5.1. The constants cL(A) and operations pL(A) are well-defined.

Proof. Let c E .c be a constant symbol from .c. Then, from the definition of the direct system of formulas, there exists i E I such that 'Pi contains a conjunct Xi = c for some Xi E Xi. Suppose that there exists another index j E I for which 'Pj contains a conjunct Xj = c, Xj E X j . Since ~ is a direct order on I then there exists k E I such.that i, j ~ k. The formula 'Pk contains conjuncts "tik(Xi) = c and "tjk(Xj) = c. Since 'Pk is realizable one has "tik(Xi) = "tjk(Xj), so (xi,i) == (Xj,j). This shows that cL(A) is well-defined.

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Let F E ,c be a functional symbol and (Xl, i l ), ... , (xnF' inF ) E

L(A). There exists jo E I such that il,"" inF :( jo, in particu­lar, l'ido(XI), ... ,l'inF jo(xnF ) E Xjo' Then by the definition of the di­rect system there exists j E I such that jo :( j and 'Pj contains a conjunct FC1'jojC1'ido(XI)), ... ,I'joj C1'inF jo (x nF )) = Xj, Xj E Xj. Since

I'jojC1'ikjO(Xk)) = l'ikj(Xk), k = 1, ... , nF, and il, ... , inF :( j then FL(A) is defined on (Xl, i l ), ... , (XnF' inF )·

Suppose there exists another i E I such that il, ... , inF :( i and 'Pi contains a conjunct FC1'it i (XI),'" ,l'inF i(XnF )) = Xi for some Xi E Xi' Then there exists k E I such that i, j :( k and 'Pk contains the conjuncts

FC1'itk(Xt}, ... ,l'inF k(XnF )) = I'jk(Xj) and FC1'i,k(XI), ... , l'inF k(XnF )) = l'ik(Xi). The diagram-formula 'Pk is consistent, hence I'idxi) = I'jk(Xj) (oth­erwise 'Pk should contain l'ik(Xi) =f. I'jk(Xj) which is impossible), therefore (xi,i) == (Xj,j).

It is left to show that the value FL(A)«(XI,il), ... , (xnF,inF )) does not depend on the representatives (Xk' ik) in the equivalence classes (Xk, ik), k = 1, ... , nF. The argument is similar to the one above and we omit it. 0

Definition 5.2. Let A = (I, 'Pi, I'ij) be a direct system of formulas in 'c. Then the set L(A) with the constants CL(A) and operations FL(A) defined above for c, F E ,C is an 'c-structure termed the limit algebra of A or a limit algebra in ,c.

Lemma 5.2. Let A = (I, 'Pi, I'ij) be a direct system of formulas in'c. Then all formulas 'Pi, i E I, hold in the limit algebra L(A) under the interpreta­tion X f----+ (x,i), X E Xi, i E I.

Proof. The result follows directly from the construction of the limit algebra L(A). 0

Lemma 5.3. Let A = (I, 'Pi, I'ij) be a direct system of formulas in .c. Suppose {13i , i E I} is a family of 'c-algebras such that the formula 'Pi can be realized in 13i , i E I. Then there is an ultrafilter D over I such that the algebra L(A) embeds into the ultraproduct DiEI 13i / D.

Proof. By the conditions of the lemma for every i E I the formula 'Pi holds in 13i under some interpretation hi : Xi -+ B i. Define a map

fo: {(x,i),x E Xi,i E I} -+ II 13i iEI

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such that fo(x, i) = b E IliEf 8 i , where b(j) = hj (-Yij (x)), if i ~ j, and b(j) = hi(x), if i > j. To define an ultrafilter D over I put Ji = {j E I, i ~ j}, i E I. Since (I, ~) is a direct ordering any finite intersection of sets from Do = {Ji liE I} contains a set from Do. Therefore, there is an ultrafilter D on I such that Do ~ D. Now, we define a map f : L(A) -) IliEf 8d D by the rule: f((x,i)) = fo(x,i)jD, (x,i) E L(A).

It is not hard to verify that the map f is well-defined and is an injective C-homomorphism. 0

Definition 5.3. Let A = (I, 'Pi, lij) be a direct system of formulas in C and 8 and C-algebra. If every 'Pi can be realized in 8 then the limit algebra L(A) is called a limit algebra over 8. In this case we denote L(A) by 8 A.

Corollary 5.1. Let A = (I, 'Pi, lij) be a direct system of formulas in C, 8 an C-algebra, and 8A a limit algebra over 8. Then there exists an ultrafilter D over I such that the limit algebra 8 A embeds into the ultrapower 8 f j D of8.

The next result explain why limit algebras over 8 have this name.

Lemma 5.4. Let C be a limit algebra over 8. Then C (viewed as a struc­ture in the relational language cre!) is the limit of a direct system of local submodels of 8.

Proof. Straightforward. o

Let X = {Xb, bE B} be a set of variables indexed by elements from B. Now let I be the set of all pairs (C' , X'), where C' is a finite reduct of the language C and X' a finite subset of X. Denote by 8 ' C'-reduct of 8 and by 'P(J:/ ,XI) the conjunction of all formulas ¢ such that (i) ¢ or -,¢ is in the core diagram Diago(8/), (ii) V(¢) ~ X', (iii) 8 F ¢ under the interpretation Xb f--+ b, Xb E X'. Clearly, 'P(CI,X/) is a diagram-formula. Conversely, every diagram-formula realizable in 8 can be obtained in the form of 'P(CI,X/).

Define (C',X/) ~ (C",X") if and only if when C' ~ C" and X' ~ X". It is easy to see that (I,~) is a direct ordering. Define the

maps I(C',X/),(C",X")' (C' , X') ~ (C", X"), as the identical maps, Le., I(CI,X/),(CII,XII)(Xb) = Xb for all Xb E X'. Straightforward verification shows that

106

is a direct system of formulas in L.

Lemma 5.5. Let B be an L-algebra and AB the direct system defined above. Then L(AB) ~ B.

Proof. Notice that for every bI ,b2 E Band i,j E I the equality (Xbl' i) == (Xb

2,j) holds in the limit algebra L(AB) if and only if bI = b2 • Therefore,

the map f : B ~ L(AB), defined by f(b) = (Xb, i) for any i E I, is a bijection. It is easy to check that f is an L-homomorphism. 0

Lemma 5.6. Let Band C be L-algebras. If Th3(B) ~ Th3(C) then C is isomorphic to some limit algebra over B.

Proof. By Lemma 5.5 L(AC) ~ C. The inclusion Th3(B) ~ Th3(C) shows that all diagram-formulas in the direct system AC are realizable in B. Hence L(AC) is a limit algebra over B. 0

5.2. Limit A-algebras

In this section we discuss limit algebras in the category of A-algebras.

Definition 5.4. Let A be an L-algebra and LA the language 12 with con­stants from A. If B is an A-algebra and A a direct system of formulas in LA then the algebra BA is called a limit A-algebra over B.

Lemma 5.7. Let A be an L-algebra, B an A-algebra, and A a direct system of formulas in LA. Then the limit algebra BA is an A-algebra, i.e., BA 1= Diag(A).

Proof. It is not hard to prove the result directly from definitions. However, it follows immediately from Corollary 5.1. 0

Since, in the notation above, the limit algebra BA is an A-algebra, all the results from Section 5.1 hold (after an obvious adjustment) in the category of A-algebras. We just mention these results without proofs.

Corollary 5.2. Let B be an A-algebra in the language LA and BA the limit

A-algebra over B relative to the direct system A = (I, 'Pi, 'Yij). Then there exists an ultrafilter D over I such that BA A-embeds into the ultrapower BI / D of the algebra B.

Corollary 5.3. Let B be an A-algebra in LA and A~ be a direct system of formulas in LA, corresponding to B (see Lemma 5.5). Then L(A~) ~A B.

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Corollary 5.4. Let A be an C-algebra, Band C A-algebras and Th3,A(B) ;;2 Th3,A(C). Then C is A-isomorphic to some A-algebra which is a limit algebra over B.

6. Unification Theorems

Theorem A [No coefficients} Let B be an equationally Noetherian algebra in a functional language C. Then for a finitely generated algebra C of C the following conditions are equivalent:

(1) Thv(B) ~ Thv(C), i.e., C E Ucl(B); (2) Th3(B) ;;2 Th3(C); (3) C embeds into an ultrapower of B; (4) C is discriminated by B; (5) C is a limit algebra over B; (6) C is defined by a complete atomic type in the theory Thv(B) in C; (7) C is the coordinate algebra of an irreducible algebraic set over B defined

by a system of coefficient-free equations.

Theorem B [With coefficients} Let A be an algebra in a functional lan­guage C and B an A-equationally Noetherian A-algebra. Then for a finitely generated A-algebra C the following conditions are equivalent:

(1) Thv,A(B) ~ Thv,A(C), i.e., C E UclA(B); (2) Th3,A(B) ;;2 Th3,A(C); (3) C A-embeds into an ultrapower of B; (4) C is A-discriminated by B; (5) C is a limit algebra over B; (6) C is an algebra defined by a complete atomic type in the theory Thv,A(B)

in the language CA; (7) C is the coordinate algebra of an irreducible algebraic set over B defined

by a system of equations with coefficients in A.

Proof. We prove here only Theorem A, the argument for Theorem B is similar and we omit it. Equivalence 1) {=} 2) is the standard result in mathematical logic.

Equivalence 1) {=} 3) has been proven in Lemma 3.8 (in the form

Ucl(B) = SPu(B)). Equivalence 1) {=} 6) has been proven in Lemma 4.3.

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To see that I} is equivalent to 5} observe first that by Corollary 5.1 one has 5) ==} 3), hence 5) ==} 1). The converse implication 1) ==} 5) follows from Lemma 5.6.

Implication 4) ==} 1) follows from Dis(B) ~ Ucl(B) (see Lemma 3.8). Now we prove the converse implication 1) ==} 4). Suppose that C tf­

Dis(B). It suffices to show that C tf- Ucl(B). Let X = {Xl, ... ,Xn } be a finite set of generators of C and (X I S) a presentation of C in the generators X, where S ~ At.c(X). The latter means that C '::0:' T.c(X)/Bs.

Since B does not discriminate C there are atomic formulas (ti = Si) E

At.c(X) , (ti = Si) tf- [S], i = 1, ... , m, such that for any homomorphism h : C ----; B there is an index i E {I, ... , m} for which h(ti/Bs) = h(si/Bs). This means that for any point p E V 6(8) there is an index i E {I, ... , m}, with t~ (p) = s~ (p). Since B is equationally Noetherian there exists a finite subsystem So ~ S such that V6(SO) = V6(S), Therefore, the following universal statement holds in B

(t=s)ESo

On the other hand the formula

(t=s)ESo

m

i=l

m

i=l

is false in C under the interpretation Yi I--' Xi, i = 1, ... , n, hence C tf-Ucl(B).

Equivalence 4) {=} 7) follows from Theorem 4.2. D

Remark 6.1. In the case when A = B the first two items in Theorem B can be formulated in a more precise form: C =v,04 A, and C =::3,04 A, correspondingly.

References

1. K. I. Appel, One-variable equations in free groups, Proc. Amer. Math. Soc., 19 (1968), pp. 912-918.

2. B. Baumslag, Residually free groups, Proc. London Math. Soc., 17 (3) (1967), pp.402-418.

3. C. Baumslag, On generalized free products, Math. Zeit., 7 (8) (1962), pp·423-438.

4· C. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups I: Algebraic sets and ideal theory, J. Algebra, 219 (1999), pp. 16-79.

109

5. G. Baumslag, A. Myasnikov, V. Remeslennikov, Discriminating and co­discriminating groups, J. Group Theory, 3 (4) (2000), pp.467-479.

6. G. Baumslag, A. Myasnikov, V. Remeslennikov, Discriminating completions of hyperbolic groups, Geometriae Dedicata, 92 (2003), pp. 115-143.

7. G. Baumslag, A. Myasnikov, V. Romankov, Two theorems about equationally Noetherian groups, J. Algebra, 194 (1997), pp. 654-664.

8. M. Bestvina, M. Feighn, Stable actions of groups on real trees, Invent. Math. J., 121 (2) (1995), pp. 287-321.

9. R. Bryant, The verbal topology of a group, J. Algebra, 48 (1977), pp. 340-346. 10. C. Champetier, V. Guirardel, Limit groups as limits of free groups: Compact­

ifying the set of free groups, Israel J. Math., 146 (2005), pp. 1-76. 11. O. Chapuis, V-free metabelian groups, J. Symbolic Logic, 62 (1997), pp. 159-

174· 12. D. Gaboriau, G. Levitt, F. Paulin, Pseudogroups of isometries of R and Rips'

theorem on free actions on R-trees, Israel J. Math., 87 (1994), pp.403-428. 13. A. Gaglione, D. Spellman, Some model theory of free groups and free algebras,

Houston J. Math., 19 (1993), pp. 327-356. 14· V. Guirardel, Limit groups and group acting freely on JRn -trees, Geometry

and Topology, 8 (2004), pp. 1427-1470. 15. V. A. Gorbunov, Algebraic theory of quasivarieties, Nauchnaya Kniga,

Novosibirsk, 1999; English transl., Plenum, 1998. 16. R. I. Grigorchuk, P. F. Kurchanov, On quadratic equations in free groups,

Contemp. Math., 131(1) (1992), pp.159-171. 17. D. Groves, Limits of (certain) CAT{O) groups, I: Compactification, Algebraic

and Geometric Topology, 5 (2005) , pp. 1325-1364. 18. D. Groves, Limit groups for relatively hyperbolic groups, II: Makanin­

Razborov diagrams, Geometry and Topology, 9 (2005), pp. 2319-2358. 19. E. Daniyarova, Foundations of algebraic geometry over Lie algebras, Herald

of Omsk University, Combinatorical methods in algebra and logic (2007), pp.8-39.

20. E. Daniyarova, I. Kazachkov, V. Remeslennikov, Algebraic geometry over free metabelian Lie algebras I: U -algberas and universal classes, J. Math. Sci., 135(5) (2006), pp.3292-3310.

21. E. Daniyarova, I. Kazachkov, V. Remeslennikov, Algebraic geometry over free metabelian Lie algebras II: Finite fields case, J. Math. Sci., 135(5) (2006), pp.3311-3326.

22. E. Daniyarova, Algebraic geometry over free metabelian Lie algebras III: Q-algebras and the coordinate algebras of algebraic sets, Preprint, Omsk, OMGU, 2005, pp.1-130.

23. E. Daniyarova, V. Remeslennikov, Bounded algebraic geometry over free Lie algebras, Algebra and Logic, 44(3) (2005), pp. 148-167.

24. O. Kharlampovich, A. Myasnikov, Irreducible affine varieties over free group I: Irreducibility of quadratic equations and Nullstellensatz, J. Algebra, 200 (2) (1998), pp.472-516.

25. O. Kharlampovich, A. Myasnikov, Irreducible affine varieties over free group II: Systems in trangular quasi-quadratic form and description of resid-

110

ually free groups, J. Algebra, 200(2) (1998), pp. 517-570. 26. O. Kharlampovich, A. Myasnikov, Algebraic geometry over free groups: Lift­

ing solutions into generic points, Contemp. Math., 378 (2005), pp. 213-318. 27. O. Kharlampovich, A. Myasnikov, Elementary theory of free nonabelian

groups, J. Algebra, 302 (2) (2006), pp.451-552. 28. R. C. Lyndon, Groups with parametric exponents, Trans. Amer. Math. Soc.,

96 (1960), pp. 518-533. 29. G. Makanin, Equations in free groups, Izvestia AN USSR, math., 46(6)

(1982), pp.1199-1273. 30. A. I. Malcev, Algebraic structures, Nauka, Moscow, 1970. 31. A. I. Malcev, Some remarks on quasi-varieties of algebraic structures, Algebra

and Logic, 5 (3) (1966), pp. 3-9. 32. D. Marker, Model theory: An introduction, Springer- Verlag New York, 2002. 33. A. Myasnikov, V. Remeslennikov, Exponential groups 2: Extension of cen­

tralizers and tensor completion of CSA-groups, International J. Algebra and Computation, 6(6) (1996), pp. 687-711.

34. A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups II: Logical foundations, J. Algebra, 234 (2000), pp.225-276.

35. A. Myasnikov, V. Remeslennikov, D. Serbin, Regular free length functions on Lyndon's free Z(t)-group p'L(t) , Contemp. Math., 378 (2005), pp. 37-77.

36. B. Plotkin, Varieties of algebras and algebraic varieties. Categories of alge­braic varieties, Siberian Advances in Math., 7 (2) (1997), pp. 64-97.

37. B. Plotkin, Varieties of algebras and algebraic varieties, Izrael J. Math., 96 (2) (1996), pp. 511-522.

38. A. Razborov, On systems of equations in a free groups, Combinatorial and ge­ometric group theory, Edinburgh (1993), Cambridge University Press (1995), pp. 269-283.

39. A. Razborov, On systems of equations in a free groups, Izvestia AN USSR, math., 48(4) (1982), pp. 779-832.

40. v. Remeslennikov, 3-free groups, Siberian Math. J., 30(6) (1989), pp.998-1001.

41. V. Remeslennikov, Dimension of algebraic sets in free metabelian groups, Fundam. and Applied Math., 7 (2000), pp. 873-885.

42. V. Remeslennikov, R. Stohr, On algebraic sets over metabelian groups, J. Group Theory, 8 (2005), pp.491-513.

43. V. Remeslennikov, R. Stohr, On the quasivariety generated by a non-cyclic free metabelian group, Algebra Colloq., 11 (2004), pp.191-214.

44· v. Remeslennikov, N. Romanovskii, Metabelian products of groups, Agebm and Logic, 43(3) (2004), pp.190-197.

45. V. Remeslennikov, N. Romanovskii, Irreducible algebmic sets in metabelian groups, Agebm and Logic, 44(5) (2005), pp.336-347.

46. V. Remeslennikov, E. Timoshenko, On topological dimension of u-groups, Siberian Math. J., 47(2) (2006), pp.341-354.

47. Z. Sela, Diophantine geometry over groups I: Makanin-Razborov diagmms, Publications Mathematiques de l'IHES, 93 (2001), pp.31-105.

48. Z. Sela, Diophantine geometry over groups VI: The elementary theory of a

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free group, GAFA, 16 (2006), pp. 707-730.

REFLECTIONS ON COMMUTATIVE TRANSITIVITY

Benjamin Fine

Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, United States

Gerhard Rosenberger

Fachbereich Mathematik, Universitat Dortmund, 44227 Dortmund, Federal Republic of Germany

Dedicated to Tony Gaglione on his 60th birthday.

Table of Contents o. Introduction 1. Commutative Transitive Groups 2. Commutative Transitivity,CSA and Universally Free Groups 3. The Commutative Transitive Kernel 4. RG Groups and the Classification of One-Relator CT Groups 5. Discriminating Groups 6. An Extension of Commutative Transitivity

1. Introduction

A group G is commutative transitive, which we will abbreviate by CT, if commutativity is transitive on nonidentity elements. Commutative tran­sitivity is a simple idea that suprisingly has had a wide-ranging impact on many areas of algebra in general and group theory in particular. Of spe­cial interest is the important role that commutative transitivity has played in the solution of the celebrated Tarski conjectures. In this article we will consider many facets of CT groups and the consequences these have had on diverse areas of finite and infinite group theory.

The term commutative transitive was coined in [F] relative to free groups and Fuchsian groups yet the concept appeared in the literature sub­stantially earlier. In some papers a CT group is referred to as centralizer

112

113

abelian or CA-group since being CT is easily shown to be equivalent to having all centralizers of nontrivial elements abelian.

Finite CT groups were studied originally by Weisner [W] in 1925. He proved that finite CT groups are either solvable or simple. However there was a mistake in his proof. Yu-Fen Wu in 1997 [Wu] corrected the mistake and reproved Weisner's result. She also proved that a finite solvable CT group is the semi direct product of its Fitting subgroup F by a fixed point free group of automorphisms of F. Earlier Suzuki [Su], in 1957, using char­acter theory proved that every nonabelian simple CT group is isomorphic to some PSL(2, 2/ ), f ~ 2. Interest in infinite CT groups arose via the ties to fully residually free groups and the related CSA property (see Section 2). A result of Gaglione and Spellman [GS] and independently Remeslennikov [R] showed that for residually free groups, being CT is equivalent to having the same universal theory as a nonabelian free group (see section 2). This result was one of the initial important steps in the solution of the Tarski conjectures (see Section 2).

The outline of this paper is as follows. In section 1 we formally define commutative transitivity and present many important standard examples. We then describe the work of Wu who gave a total structure theory for the class of locally finite CT groups. Finally we consider how CT groups behave under group amalgams. In section 2 we look at the CSA property and the relation to fully residually free groups, universal freeness and the Tarski con­jecture. In an effort to further understand commutative transitivity, Fine, Gaglione, Rosenberger and Spellman [FGRS 1] developed the commuta­tive transitive kernel. This is a subgroup that carries the information about commutative transitivity in much the same way that the derived group G' carries the information about commutativity. Further work on the CT kernel was done by Delizia and Nicotera (see [DN 1,2,3,4,5]). In section 3 we describe this subgroup and what is known of it. In section 4 we introduce the concept of a restricted Gromov or RG-group. We then use this property to present a classification of one-relator CT groups. In section 5 we discuss the effect of commutative transitivity on the study of discriminating groups. The class of discriminating groups (see section 5) was introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov [BMR 1, 2] as an outgrowth of their theory of algebraic geometry over groups. However discriminating groups have taken on a life of their own and have been an object of a considerable amount of study (see [FGMS]). Finally in section 6 we present an extension of coomutative transitivity,

114

2. Commutative Transitivity and Commutative Transitive Groups

Definition 2.1. The group G is commutative transitive or CT if com­mutativity is transitive on nontrivial elements. That is

[x, y] = 1 and [y, z] = 1 ==} [x, z] = 1

provided x, y, z are nontrivial.

It is straightforward that being commutative transitive is equivalent to the property that the centralizer of every nontrivial element is abelian. For this reason CT groups are sometimes called CA groups or centralizer­abelian groups.

Further we observe that the group G is CT if and only if it satisfies the universal sentence

\Ix, y, z«(y -I- 1) II (xy = yx) II (yz = zy)) -t (xz = zx)).

That commutative transitivity is defined in terms of a universal sentence becomes important in both the study of fully residually free groups and the study of discriminating groups.

It is also clear is that if Z (G) -I- {I} and G is CT then G is abelian. Harrison [H] first published the following lemma that ties together the

CT property with abelian centralizers.

Lemma 2.1 (H). Let G be a group. The following three statements are pairwise equivalent.

(i) G is commutative transitive. (ii) The centralizer C (x) of every nontrivial element in x EGis abelian. (iii) Every pair of distinct maximal abelian subgroups in G has trivial

intersection.

As mentioned in the introduction, finite CT groups were studied origi­nally by Weisner [W] in 1925 who proved that finite CT groups are either solvable or simple. However there was a mistake in his proof that was cor­rected by Yu-Fen Wu in 1997 [Wu]. She further proved that a finite solvable CT group is the semi direct product of its Fitting subgroup F, which must be abelian, by a fixed point free group of automorphisms of F. Suzuki [Su] in 1957 using character theory proved that every nonabelian simple CT group is isomorphic to some PSL(2, 21), f ~ 2.

115

Wu did extensive work on CT groups and in particular developed a complete structure theory for locally finite CT groups analogous to that of finite CT groups. Two of her main results were.

Theorem 2.1 (Wu). If G is a solvable locally finite CT group then G is the semi direct product of H by F where F is the Fitting subgroup of G which

is abelian and H is a locally cyclic group of fixed point free automorphisms of F. Conversely if F is an abelian locally finite group and H is a locally

cyclic group of fixed point free automorphisms of F, then the semidirect product G of H by F is a solvable locally finite CT group.

Theorem 2.2 (Wu). An insolvable locally finite CT group is CT if and

only if G == PSL(2, F) for some locally finite field F of characteristic 2 with IFI ~ 4,

Among non-locally finite groups, examples of CT group abound. The following important classes of groups all consist of commutative transitive groups.

(1) Free groups: It is well known that centralizers of elements in nonabelian free groups are all cyclic (see [MKS] or [LSD. This is usually expressed by saying that two element commute only if they are powers of a common element.

(2) Torsion-free hyperbolic groups: Again here centralizers are cyclic (see [FGMS1]).

(3) Free solvable groups: This was proved by Wu in the paper cited above [Wu]. She also proved that there are solvable CT groups of any de­rived length.

(4) Free met abelian groups: Again see [Wu].

(5) PSL(2, K) where K is a field with char(K) =I- 2: The prototype example is where K = C the complex numbers. Here two elements T, U E

PSL(2, q commute if and only if they have the same set of fixed points when considered as projective linear transformations from C to C. From this observation it is clear that PSL(2, q is CT. An identical argument can be applied to any PSL(2, K) where K is an infinite field of characteristic not equal to 2.

Further from the observation that PSL(2, q is CT and the obvious fact that all subgroups of CT groups are also CT it follows that all Fuchsian groups and all Kleinian groups are CT. In particular any orient able

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surface group 89 with 9 ~ 2 must be CT. This is important relative to the tie between CT groups and residually free groups (see the next section).

(6) Tarski groups: These are simple groups where every proper sub­group is cyclic hence centralizers are cyclic.

(7) One-relator groups with torsion: This is a consequence of results of Pride [P] on two-generator one-relator groups coupled with earlier results of B.B. Newman [Ne] and Ree and Mendelsohn [RM].

We mention also that there has been work by Rattagi and others (see [Raj) on commutative transitivity in Lie Groups and groups of quaternions as well as commutative transitivity in more general algebraic structures such as free Lie algebras and free associative algebras.

Centralizers of elements and abelian subgroups in general are relatively well understood relative to group amalgams, that is free products with amalgamation and HNN groups. Using these ideas, commutative transitiv­ity was studied relative to certain group amalgams by Levin and Rosen­berger [LR]. More generally they call a group commutation transitive if commutativity is transitive on noncentral elements. For centerless groups this is equivalent to CT. They also consider in the same manner power commutative groups, that is groups G where for elements a, bEG of infinite order, powers of a, b commute only if a, b commute. Recall that a lA-group is a group where extraction of roots is unique, that is xn = yn

implies that x = y for n ~ 1. In [Ku] it was shown that a torsion-free group is a lA-group if and only if it is power commutative. The main result of Levin and Rosenberger on generalized free products is the following.

Theorem 2.3 (LR). If G1,G2 are CT gmups and H is malnormal and pmper in both then the amalgamated pmduct G 1 *H G2 is also CT.

They note that the malnormality condition cannot be relaxed. For ex­ample if the values p, q > 1 then the amalgamated product

< a > *K < b, c; [b, c] = 1 > where K =< aP >=< bq > is not CT.

The situation for HNN extensions of CT groups is not as general but can be carried through for extensions of centralizers of abelian malnormal subgroups.

Theorem 2.4. Let B be a CT gmup and K an abelian malnormal subgmup of B. Then the HNN extension

Bl =< t,B;rel(B),r1kt = k for all k E K >

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is aCT-group.

3. Commutative Transitivity, CSA and Universally Free Groups

Myasnikov and Remeslennikov in their study of fully residually free groups introduced the concept of a CSA group (conjugately separated abelian group). First we recall the following necessary definition.

Definition 3.1. Let G be a group and H a subgroup of G. His malnor­mal in G or conj ugately separated in G provided g -1 H g n H = 1 unless g E H.

Using this we define the concept of a CSA group.

Definition 3.2. A group G is a CSA-group or conjugately separated abelian group provided the maximal abelian subgroups are malnormal.

Each CSA group must be CT. The converse however is not true in general.

Lemma 3.1. The class of CSA groups is a proper subclass of the class of CT groups.

Proof. We first show that every CSA-group is commutative transitive. Let G be a group in which maximal abelian subgroups are malnormal and suppose that M1 and M2 are maximal abelian subgroups in G with z =J 1 lying in M1 n M 2. Could we have M1 =J M2? Suppose that w E M1 \ M 2. Then w- 1 zw = z is a non-trivial element of w-1 M 2w n M2 so that w E

M 2. This is impossible and therefore M1 c M2 . By maximality we then get M1 = M 2. Hence, G is commutative transitive whenever all maximal abelian subgroups are malnormal.

We now show that there do exist CT groups that are not eSA. In any non-abelian CSA-group the only abelian normal subgroup is the trivial subgroup 1. To see this suppose that N is any normal abelian subgroup of the non-abelian CSA-group G. Then N is contained in a maximal abelian subgroup M . Let g ~ M . Then

N = g-lNgn N c g-lMg n M.

The fact N =J 1 would imply that gEM which is a contradiction. Now let p and q be distinct primes with p a divisor of q - 1. Let G be

the non-abelian group of order pq . Then it is not difficult to prove that

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the centralizer of every non-trivial element of G is cyclic of order either p

or q . Thus G is commutative transitive. However, the (necessarily unique) Sylow q-subgroup of G is normal in G. Hence from the argument above G cannot be CSA. 0

Although the class of CSA groups is a proper subclass of the CT groups in the presence of full residual freeness (in fact even in the presence of just residual freeness) they are equivalent.

Definition 3.3. A group G is residually free if for each non-trivial 9 E G there is a free group Fg and an epimorphism hg : G ---; Fg such that hg(g) #- 1. Equivalently for each 9 E G there is a normal subgroup Ng such that G/Ng is free and 9 rJ. Ng •

The group G is fully residually free provided to every finite set S c G \ {1} of non-trivial elements of G there is a free group Fs and an epimorphism hs : G ---; Fs such that hs(g) #- 1 for all 9 E S.

Lemma 3.2. If G is fully residually free then GT == GSA.

The proof of this depends on a beautiful theorm due independently to Gaglione and Spellman [GS] and Remeslennikov [R] tying together full residual freeness and the property of being universally free which we will explain shortly (see [R] for a proof of Lemma 2.3).

In the 1960's G. Baumslag [GB 1] proved that a surface group is resid­ually free, answering a question of Magnus. To do this he introduced what is now called extensions of centralizers. This concept became one of the main tools used by Kharlampovich,Mayasnikov and Remeslennikov in their structure theory of fully residually free groups and by Kharlampovich and Maysnikov in their solution to the Tarski problems. As somewhat of an offshoot of this B. Baumslag [BB 1] proved:

Theorem 3.1. If G is residually free then the following are equivalent: (1) G is fully residually free (2) G is GT

From this a truly amazing result developed that is in some sense the be­ginning of the solution of the Tarski problem. This was done independently by Gaglione and Spellman and Remeslennikov: Recall first some ideas from logic and model theory.

We start with a first-order language appropriate for group theory. This language which we denote by Lo is the first-order language with equality

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containing a binary operation symbol . a unary operation symbol -1 and a constant symbol 1. A sentence in this language is a logical expression containing a string of variables x = (Xl, ... , x n ), the logical connectives V, /\, rv and the quantifiers V, 3. A universal sentence of Lo is one of the form \fX{ ¢(x)} where x is a tuple of distinct variables, ¢(x) is a formula of Lo containing no quantifiers and containing at most the variables of X.

Similarly an existential sentence is one of the form 3x{ ¢(x)} where x and ¢(x) are as above.

If G is a group then the universal theory of G consists of the set of all universal sentences of Lo true in G. We denote the universal theory of a group G by T hv( G). Since any universal sentence is equivalent to the negation of an existential sentence it follows that two groups have the same universal theory if and only if they have the same existential theory. The set of all sentences of Lo true in G is called the first-order theory or the elementary theory of G. We denote this by Th(G). We note that being first-order or elementary means that in the intended interpretation of any formula or sentence all of the variables (free or bound) are assumed to take on as values only individual group elements - never, for example, subsets of, nor functions on, the group in which they are interpreted. The Tarski conjectures, solved independently by Kharlampovich and Myasnikov (see [KHM 1-5]) and Sela (see [Se 1-5]), say essentially that all countable nonabelian free groups have the same elementary theory. The following was well-known and much simpler.

Theorem 3.2. All nonabelian free groups have the same universal theory.

As we will see all finitely generated fully residually free groups will also have the same universal theory as the class of nonabelian free groups.

Definition 3.4. A universally free group G is a group that has the same universal theory as a nonabelian free group.

Gaglione and Spellman [GS] and independently Remeslennikov [Re] were able to extend the theorem of B. Baumslag to show that fully resid­ually free is equivalent to universally free and that these (in the presence of residual freeness) are equivalent to both being CT and being CSA. As mentioned this is in some ssnse the starting off point for the solution of the

Tarksi conjectures.

Theorem 3.3 (GS). ,[Re) If G is residually free then the following are

equivalent:

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(1) G is fully residually free (2) G is GT (3) G is GSA (4) G is universally free

Since this result a complete structure theory and algorithmic theory of the fully residually free groups has been developed. An important aspect of this development is that elements of fully residually free groups can be expressed as infinite words on a generating system. These infinite words can be manipulated and handled in an analogous manner to ordinary words in free groups (see [KMRS]).

Commutative transitivity becomes essential in building examples of fully residually free groups classifying them via the following construction.

Definition 3.5. Let G be a CT group, let u E G\ {I} and let M = Za(u)

where Za(u) is the centralizer of u in G. Suppose A is an abelian group. Then the group

H =< G,A; reI (G), reI A, [A,z] = 1,Vz EM>

is a centralizer extension of G by A. If A =< t > is cyclic then H = G(u, t) is the HNN extension

G(u,t) =< G,t; reI (G),CIzt = z, for all z EM>

and is called the free rank one extension of the centralizer M of u in G.

Theorem 3.4. (Baumslag, Myasnikov, Remeslennikov [BMR3j) Let G be a fully residually free group and A an abelian fully residually free group. Then a centralizer extension of G by A is again fully residually free.

The proof of this result which is fundamental in all further consider­ations of fully residually free groups depends on the fact that the result can be reduced to free rank one extensions of centralizers and then on the following "big powers" argument. It is not hard to see that in a free group F if botn'bl ... tnkbk = 1 for infinitely many values of nl, infinitely many values of n2"" infinitely many values of nk then t must commute with at least one of bo, ... , bk. Hence the family of homomorphisms ¢k : F(u, t) ----t F from the rank one extension of the centralizer G F (u) into F, defined for every positive k by ¢(t) = uk and ¢k IF= id, is a discriminating family, as required.

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As mentioned earlier, G.Baumslag [GB 1] used this type of argument to show that the orient able surface groups 8g with g 2: 2 are all residually free. This answered a question posed by Magnus. Recall that for g 2: 2 the group 8g which is the fundamental group of an orient able surface of genus g has the presentation

8g =< aI, bl , ... , ag, bgi [aI, bl] ... [ag, bg] = 1 > .

Baumslag observed that each 8g embeds in 82 and residual freeness is inherited by subgroups so it suffices to show that 82 is residually free. He actually showed more. If F is a nonabelian free group and u E F is a nontrivial element which is neither primitive nor a proper power then the group K given by

K =< F * Fi U = U >

where F is an identical copy of F and u is the corresponding element to u in F, is residually free. A one-relator group of this form is called a Baumslag double. In our terminology he proceeded by embedding K in the free rank one extension of centralizers

H =< F,tiClut = u >

by

K =< F,CIFt >.

The group H is then residually free and hence K is residually free. Therefore every Baumslag double is residually free. The group

82 =< al,bl ,a2,b2i[al,bl ] = [a2,b2] >

is a Baumslag double answering the original question. The class of finitely generated fully residually free groups was introduced

in a different direction by Sela in his proof of the Tarski problems. In Sela's approach these groups appear as limits of homomorphisms of a group G into a free group. In this guise they are called limit groups. Therefore a limit group is a finitely generated fully residually free group. The paper by Bestvinna and Feign [BeF] gives a nice description of the equivalence of the two approaches.

4. The Commutative Transitive Kernel

Commutative transitivity and universal freeness raised certain questions about the relationship between fully residually free and n-free.

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Definition 4.1. A group G is n-free if any subset of n or fewer elements in G generate a free group.

Observe that if G is an n-free group then G is also an m-free group for all 1 :s; m < n. The I-free groups are precisely the torsion free groups. 2-free groups are commutative transitive.

Lemma 4.1. If G is an n-free group for any n ~ 2 then G is commutative transitive.

Proof. Suppose that G -I- 1 is a 2-free group and let u E G. Let M = Ga ( u) . We claim that M is locally cyclic and therefore abelian. Suppose a, b E

M. Since G is 2-free, a, u generate a free group. Since a, u commute this must be cyclic and hence they are both powers of a single element g. Thus a = gCt, U = g{3 for some Ct, (3. Similarly, since band u commute and b, u generate a free group, we have an element h with b = h ii

, U = h'Y for some 8, 'Y. Now consider < h, g >c G. This is free because G is 2-free however this has the relation g{3 = h'Y. Therefore < h, 9 > must be cyclic and hand 9 are both powers of a single element. Hence a and b are both powers of this element and any 2-generator subgroup of M is cyclic. A straightforward induction then shows that any n-generator subgroup < aI, ... , an >C Mis also cyclic. Thus, centralizers of non-trivial elements are locally cyclic and 2-free groups are commutative transitive as claimed. 0

The concept of n-freeness arises in many places. From straightforward topological considerations it follows that an orientable surface group of genus 9 ~ 1 is 2g - 1 free while a nonorientable surface gorup of genus 9 ~ 2 is g - 1 free. These results were generalized in an algebraic manner by Fine, Gaglione,Rosenberger and Spellman (see [FGRS 1]).

In order to further study commutative transitivity, Fine, Gaglione, Rosenberger and Spellman [FGRS 2] introduced a subgroup T( G) called the commutative transitive kernel that carries the information about commutative transitivity in much the same way that the derived group G' carries the information about commutativity.

Definition 4.2. Let N be a normal subgroup of G and let To(G, N) = N and inductively define

Tn+I(G, N) = subgroup of G generated by Tn(G, N) tgether with those commutators [a, c] such that aRnc where Rn is the transitive closure of the relation on G - Tn(G, N) by xy == yx mod Tn(G, N). Now let

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Then T(G,N) eNG' and T(G,N) is normal in G

Lemma 4.2. Let N be normal in G. Then GjN is CT if and only if N = T(G, N). Further GjT(G, N) is CT

Definition 4.3. The commutative transitive kernel of a group G, de­noted T(G), is T(G, 1).

The commutative transitive kernel behaves in much the same way as the derived group. In particular we have the following theorem.

The important result is

Theorem 4.1. Let G be a group. Then (1) T( G) is a characteristic subgroup of G and contained in G' (2) GjT(G) is CT (3) Gis CT if and only ifT(G) = 1

Unfortunately the commutative transitive kernel is not canonical for commutative transitivity in the same sense as the derived group is for com­mutativity. To make this precise let F be a covariant functor on the category of groups. We say that F is a T-like functor if for each group G

[lJ F(G) is a characteristic subgroup f G and is contained in G' [2J G j F( G) is commutative transitive [3J G is commutative transitive if and only if F(G) = {I}.

The set of T-like functors can be partially ordered by Fl :::; F2 if Fl (G) c F2(G) for every group G. A canonical subgroup for commutative transitivity would be equivalent to a minimum T-like functor. In [FGRS 1 J the following was proved.

Theorem 4.2. The family of T-like functors has no minimum.

C. Delizia and C. Nicotera further studied the commutative transitive kernel for locally finite groups [DN 1,2J. They also developed an analog of the commutative transitive kernel for power commutative groups [DN 3,4J and study the structure of this kernel for locally nilpotent groups. In another direction they consider groups where nilpotency is transitive on 2-generator subgroups [DN 5J.

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5. RG Groups and a Classification of One-Relator CT Groups

In presenting examples of infinite CT groups we mentioned that one-relator groups with torsion are CT. The question then arises as to how to classify both the one-relator CT groups and the one-relator CSA groups. In this section we describe a recent classification of such one-relator groups done by Fine, grose Rebel, Myasnikov and Rosenberger [FgRMR].

This classification is related to two more general questions. (1) The general classification of one-relator fully residually free

groups (2) The Gersten conjecture that a torsion-free one-relator group is

hyperbolic if and only if it does not contain any Baumslag-Solitar group

B m -1 n --I- 0 m,n =< x, y; yx y = x >, mn r .

Recall that one-relator groups with torsion are hyperbolic. In order to present the classification we need another concept introduced

by Fine and Rosenberger [FR] and independently by M.Cohen and M.Lustig [CL]

Definition 5.1. A group G is a restricted Gromov or RG group if given g, h E G either < g, h > is cyclic or there exists a positive integer t with gf -1= 1,ht -1= 1 and < gt,ht >=< gf > * < ht >.

In particular free groups and torsion-free hyperbolic groups are RG.

Theorem 5.1 (FR). Torsion-free RG groups are CT.

Now the classification. For one-relator groups with torsion, combining results of Fine and Rosenberger [FR] and Gildenhuys,Kharlampovich and Myasnikov [GKM] we obtain the following result ([FgRMR]).

Theorem 5.2. ((FgRMRJ) Let G be a one-relator group with torsion. Then the following are equivalent:

(1) Gis CSA (2) Gis RG (3) G does not contain a copy of the infinite dihedral group <

x, y; x2 = y2 = 1 >

For torsion-free one-relator groups the situation is different. A torsion­free one-relator group fails to be a CSA group if and only if its contains

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a copy of some nonabelian Baumslag-Solitar group BI,n with n -I- 1 or a copy of the group FI x Z, the direct product of a free group of rank 2 and an infinite cyclic group (see [FgRMR]). In [FgRMR] it was proved that a one-relator group fails to be an RG-group if and only if it contains a copy of one of the Baumslag-Solitar groups BI,n. Recall that BI,1 is a free abelain group of rank 2. It follows that if G is a torsion-free one-relator group which does not contain a free abelian subgroup of rank 2 then the following are equivalent.

(1) G is a CSA group (2) G is an RG-group (3) G does not contain a copy of some BI,n with n -I- -1,0,1.

Further if G is a torsion-free one-relator group then G is CT if and only if it does not contain a copy of F2 x Z or a copy of the Kleian bottle group BI,-l. Combining all these we get the following results.

Theorem 5.3. ([FgRMRj) Let G be a torsion-free one-relator group which does not contain a copy of Z x Z = BI,I. Then the following are equivalent:

(1) G is GSA (2) Gis RG

(3) G does not contain a copy of one of the Baumslag-Solitar group BI,m =< x, y : yxy-l = xm > with mE Z \ {-I, 0, I}.

Theorem 5.4. ([FgRMRj) Let G be a torsion-free one-relator group. Then G is GT if and only if G does not contain a copy of F2 X Z or a copy of the Baumslag-Solitar group BI,-l =< x, y; yxy-l = X-I > (the Klein-bottle group).

Notice that since one-relator groups with torsion are commutative tran­sitive, this fact together with Theorem 4.3 provides the total classification of one-relator commutative transitive groups.

6. Commutative Transitivity and Discriminating Groups

As an outgrowth of the development of the theory of algebraic geometry over groups (see [BMR2] [BMR1]), Baumslag, Myasnikov and Remeslen­nikov introduced the concept of discriminating groups. This class of groups developed a life of their own and have been extensively studied (see [FGMRS] and the references there).

Definition 6.1. Let G and H be groups. G separates H provided that to every nontrivial element h of H there is a homomorphism 'Ph : H -+ G

126

such that CPh(h) -=I- 1. G discriminates H if to every finite nonempty set S of nontrivial elements of H there is a homomorphism CPS : H ----+ G such that cps(s) -=I- 1 for all s E S. The group G is discriminating provided that it discriminates every group it separates.

Algebraic geometry over groups was created as a tool to attack the celebrated Tarski conjectures on the elementary theory of free groups. By analogy with classical algebraic geometry we may view the discrimination of H by G as an approximation to H much like the localization of a ring at a prime. (Think of a set of generators for H as a set of variables.)

The following is the main criterion for determining whether a group is discriminating.

Lemma 6.1. ([BMRj) A gmup G is discriminating if and only if G dis­criminates G x G.

In the application of algebraic geometry it is as important to know both whether a group is discriminating and whether it is not discrimninating. Hence there are general questions on nondiscrimination of various classes of groups.

General idea to show nondiscrimination is to show that some universal property which is true in G but cannot be true in G x G or to find a number - dimension etc. - which is additive so that this number cannot hold in both G x G and in G. Here commutative transitivity plays a large role.

Lemma 6.2. A nonabelian CT gmup is nondiscriminating.

It follows that all the following classes of groups are nondiscriminating: (1) Any torsion-free hyperbolic group and in particular any non­

abelian free group is non discriminating

(2) Nonabelian free solvable groups and their nonabelian sub­groups are nondiscriminating.

(3) The free product of two nondiscriminating groups is nondis­criminating.

In subsequent work it was shown that a nilpotent group is discriminating if and only if its torsion-free abelian (see [FGMS]).

7. An Extension of Commutative Thansitivity

In [BFGRS] a study was initiated to extend the concept of commutative transitivity. This begins in the following direction.

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Definition 7.1. Let V be a variety of groups that contains the abelian variety A. Then we say that a group G is a centralizer-V group or ev­group if for each 9 E G the centralizer of 9 is in V.

In this context CT-groups are centralizer abelian or CA - groups. Since varieties are closed under subgroups is is clear that any group G in V is a eV-group. Further since we are assuming that the variety V contains the abelian variety it follows that any CT-group is ev. There are certainly eV-groups that are not CT-groups. For example the group

G =< a, b; aba-1 = b- 1 >

is not CT but is eM where M is a variety which contains the abelian and metabelian variety.

We now show that there are nontrivial examples of eV-groups, that is groups G that are not in V or commutative transitive but which are ev. We call a group a nontrivial eV-group if it is ev and not in V and not CT.

Theorem 7.1. A free product of nontrivial eV-groups is a nontrivial ev­group.

Theorem 7.2. Let G and H be nontrivial eV-groups. Then the unre­stricted wreath product Gwr H is a nontrivial ev -group.

Mimicing methods of Levin and Rosenberger [LR] We can show that the class of eV-groups is closed under certain types of group amalgams.

Theorem 7.3. Let A and B be eV-groups and An B = K proper and malnormal in both A and B. Then the amalgamated free product A *K B is a eV-group. In particular a nontrivial free product of eV-groups is again ev.

Theorem 7.4. Let B be a eV-group and K a nontrivial abelian malnormal subgroup of B. Then the HNN group

Bl =< t,B;C1kt = k for all k E K >

is a eV-group.

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8. References

[AAR] P. Ackermann, V. grosse Rebel and G. Rosenberger, On Power and Commutation Transitive, Power Commutative and Restricted Gromov Groups Cont. Math. ,360,2004,1-4

[GB 1] G. Baumslag On generalized free products Math. Z., 78, 1962,423-438

[BFGS] G.Baumslag, B. Fine, A.M. Gaglione and D. Spellman, Reflections on discriminating groups, J. Group Theory, 10, 2007, 87-99

[BFGRS] G.Baumslag, B. Fine, A.M. Gaglione,G.Rosenberger and D. Spell­man, On Centralizer Varietal Groups in preparation

[BMR1] G. Baumslag, A.G. Myasnikov, V.N. Remeslennikov, Algebraic ge­ometry over groups I. Algebraic sets and ideal theory J. Algebra, 219, 1999, 16-79

[BMR2] G. Baumslag, A.G. Myasnikov, V.N. Remeslennikov, Discriminat­ing and co-discriminating groups J. of Group Theory, 3, 2000,467-47

[CL] M. Cohen and M. Lustig, Very small group actions on JR-trees and Dehn twist automorphisms Topology, 34, 1995, 575-617

[DN 1] C. Delizia and C.Nicotera, On the Commutative Transitive Kernel of Locally Finite Groups Alg. Colloquium, 10, 2003, 567-570

[DN 2] C. Delizia and C.Nicotera, On the Commutative Transitive Kernel of Certain Infinite Groups JP Journal of Algebra, Number Theory and Applications, 5, 2005, 421-427

[DN 3] C. Delizia and C.Nicotera, On the Power-Commutative Kernel of Locally Nilpotent Groups Int. J. of Math. and Math. Sciences, 17, 2005, 2719-2722

[DN 4] C. Delizia and C.Nicotera, On the Power-Commutative Kernel of Finite Groups Journal of Science, Ferdowsi University of Mashad , 5, 2005, 3-8

[DN 5] C. Delizia and C.Nicotera, Groups in Which the Bounded Nilpotency of Two-Generator Subgroups is Transitive preprint

[F] B.Fine, On Power Conjugacy and SQ-universality for Fuchsian and Kleinian Groups, in Modular Functions in Analysis and Number Theory, University of Pittsburgh Press, 1983, 41-55

[FGMS] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, Dis­criminating groups: A Comprehensive Overview CRM Preprint , 2006

129

[FGMS1] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, Dis­criminating groups J. of Group Theory, 4, 2001, 463-474

[FGRS 1] B.Fine,A.M.Gaglione,G.Rosenberger and D. Spellman, The Com­mutative Transitive Kernel Algebra Colloquium, 2, 1997, 141-152

[FGRS 2] B.Fine,A.M.Gaglione,G.Rosenberger and D. Spellman, Free Groups and Questions About Universally Free Groups, in Proceedings of Groups St. Andrews/Galway 1993, London Mathematical Soc. Lecture Notes Series 211, 1993, 191-204

[FR 1] B.Fine and G.Rosenberger, On Restricted Gromov Groups Comm.in Aig. , 20, 8, 1992, 2171-2182

[FgRMR] B.Fine, V. grosse Rebel, A. Myasnikov and G. Rosenberger, A Classification of CSA, Commutative Transitive and Restricted Gromov One-Relator Groups Result. Math. to appear

[GS] A.M. Gaglione and D. Spellman, Even More Model Theory of Free Groups, in Infinite Groups and Group Rings, World Scientific, 1993, 37-40

[GKM] D.Gildenhuys, O. Kharlampovich and A. Myasnikov, CSA Groups and Separated Free Constructions Bull. Austral. Math. Soc. ,52, 1995, 63-84

[K] M.Kasabov On Discriminating Solvable Groups preprint

[LR] F. Levin and G. Rosenberger, On Power Commutative and Commuta­tion Transitive Groups, in Proc. Groups St Andrews 1985 , Cambridge University Press, 1986 , 249-253

[LS] R.C.Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag 1977

[MKS] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Wiley 1966, Second Edition, Dover Publications, New York 1976

[MRe] A.G. Myasnikov and V.N. Remeslennikov Exponential groups 1: Foundations of the theory and tensor completion Omsk University, N9 1993 1-25

[MRe 1] A.G. Myasnikov and V.N. Remeslennikov, Exponential groups, preprint New York, 1994

[MRe 2] A.G. Myasnikov and V.N. Remeslennikov, Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups, preprint

[MS] A.M. Myasnikov and P. Shumyatsky, On Discriminating Groups and c-Dimension J. of Group Theory, 200

130

[Ne] B.B.Newman, Some results on One-Relator Groups Bull. Amer. Math. Soc. , 74, 1968, 568-571

[P] S.J. Pride, The two generator subgroups of one-relator groups with torsion Trans. Amer. Math. Soc. , 234, 1977, 483-496

[Su] M. Suzuki, The Nonexistence of a Certain Type of Simple Groups of Odd Order Proc. Amer. Math. Soc. ,8,1925,686-695

[W] L.Weisner, Groups in which the normalizer of every element but the identity is abelian Bull. Amer. Math. Soc. , 31, 1925, 413-416

[Wu] Y.F. Wu Groups in which Commutativity is a Transitive Relation J. of Algebra, 207, 1998, 165-181

GROUPS UNIVERSALLY EQUIVALENT TO FREE BURNSIDE GROUPS OF PRIME EXPONENT AND A

QUESTION OF PHILIP HALL

A. Gaglione

Department of Mathematics U.S. Naval Academy

Annapolis, MD 21402

S. Lipschutz

Department of Mathematics Temple University

Philadelphia, PA 19122

D. Spellman

Department of Mathematics Temple University

Philadelphia, PA 19122

Subject Classification: Primary 20E26; Secondary 20E05, 20E06

Abstract :This paper proposes the Tarski Problem for free groups in a Burn­side variety, B n , where n is a sufficiently large odd integer so that Adian's results hold. We note that just as in the case of absolutely free groups it is easy to show that the nonabelian free groups in Bn for n as above are all universlly equivalent.

1. Introduction

We bear in mind two problems which resisted solution for decades and succumbed only to the fresh attacks of supremely talented mathematicians. For our purposes we shall dub the questions (1) and (2) below as The Tarski Problem and The Burnside Problem respectively.

(1) Are the nonabelian free groups elementarily equivalent? (2) Must the finitely generated nonabelian free Burnside groups of fixed

finite exponent be finite?

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Of course the answers to (1) and (2) are now known to be (1) yes and (2) no respectively. (1) was solved independently by Kharlampovich and Myasnikov on the one hand and by Sela on the other. Kharlampov­ich and Myasnikov applied algebraic geometry over groups invented by G. Baumslag, Myasnikov and Remeslennikov while Sela invented Diophantine geometry in groups to solve the Tarski problem.

Let's focus on algebraic geometry over groups. By way of analogy let us consider a Noetherian integral domain R. The closed subsets in the Zariski topology on affine n-space over R are precisely the affine algebraic subsets. In order for the analog of this scenario to go through in groups the proper notions of domain and Noetherian had to be defined. One consequence of the correct definition of domain is that every nonabelian eSA group (See Section 3) is a domain. Since free groups are eSA, nonabelian free groups are domains. The correct notion of Noetherian is equationally Noetherian (See Section 6). Happily free groups are also equationally Noetherian. One felicitous consequence of this confluence of facts it, the existence and unique­ness (in the usual sense) of the decomposition of a closed set into a finite union of irreducible affine algebraic subsets. (We need to reserve the word variety for an equational class in this paper.) While the groups free in the variety of all groups are elementarily equivalent (provided their rank ex­ceeds 1) it is easy to see that the corresponding result is false for groups free in many other varieties. For example, one can distinguish the free nilpo­tent groups (of fixed class) of different finite ranks by first order sentences. Perhaps the variety of all groups is unique in this regard? Or is it!

We now turn our attention to Question (2). A negative answer was first provided by Adian who showed that, for all sufficiently large odd n, the nonabelian free groups in the variety Bn determined by the law xn = 1 were all infinite. Soon afterward Sirvanjan proved that, just as in the variety of all groups, the free group of countably infinite rank in the variety Bn embeds in its group free of rank 2 for all sufficiently large odd n. From this it easily follows that the nonabelian free groups in the varieties Bn, for all sufficiently large fixed odd n, have the same universal theory in the sense of first order logic. For our purposes we can say the most when n = p is a sufficiently large prime. In any event, if n is a sufficiently large odd integer, the free groups in Bn are eSA; hence, the nonabelian free groups in such varieties are domains. We do not know whether or not they are equationally Noetherian. The Tarski problem relativized to such Bn will require new techniques for its solution. This paper provides some halting first steps and should be viewed as an invitation to our colleagues to ponder

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possible approaches.

2. Preliminaries

We let w be the first limit ordinal, which we identify with the first infinite cardinal ~a. If Hand G are groups we say that G is H -inclusive provided it contains a subgroup isomorphic to H; G is H -exclusive provided it is not H-inclusive. For each positive integer n, en shall be a group cyclic of order n. If G is a group and H is a nonempty class of groups, then we say that H separates G provided for every 9 E G\ {I} there is a group Hg E H and a homomorphism <Pg : G -> Hg such that <pg(g) =f=. 1; we say that H discriminates G provided for every finite nonempty subset S <;;;; G\ {I} there is a group Hs E H and a homomorphism ips : G -> Hs such that <P s (g) =f=. 1 for all 9 E S. In the case H = {H} is a singleton we say that H separates (discriminates) G for H separates (discriminates) G. In the case that H separates (discriminates) G by epimorphisms the language G is residually (fully residually) H is sometimes used.

Let La be the first order language with equality containing a binary operation symbol., a unary operation symbol -1 and a constant symbol 1.

We remark that being first order means that the variables are interpreted as varying over individual elements of the domain of discourse - never over subsets nor functions. Thus an La-structure is a set G provided with a distinguished constant 1 E G, a unary operation G -> G, 9 f---+ g-l and a binary operation G2 -> G, (g, h) f---+ gh. A universal sentence of La is one of the form 'v'x<p(x) where x is a tuple of distinct variables and <p(x) is a formula of La containing no quantifiers and containing free at most the variables in x. A law in La is a universal sentence of the form 'v'x(s(x) = t(x)) where x is a tuple of distinct variables and s(x) and t(x) are terms of La containing at most the variables in x. The model class of a set of laws of La is a variety of La-structures. The following three sentences are laws.

-1'1: 'v'X1, X2, X3( (Xl. X2) • X3 = Xl • (X2 • X3)) Associative Law 1'2: 'v'x(x.1 = x) Identity Law 1'3: 'v'x(x. x-1 = 1) Inverse Law We shall refer to the set b1' 1'2,1'3} as the group axioms. The model

class 0 of the group axioms is the variety of all groups. Every variety under consideration in this paper shall be a subvariety of O. If I' is the conjunction 1'1 A 1'2 A 1'3 of the group axioms and (j and T are sentences of La then we shall say that (j and T are equivalent modulo the group axioms provided the sentence I' -> ((j f-t T) is true. Every law in La is equivalent modulo the group axioms to one of the form'v'x(w(x) = 1) where x is a

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tuple of distinct variables and w(x) is a word in at most the variables in x. Henceforth we shall omit the universal quantifiers and abbreviate the law I;fx(w(x) = 1) as simply w(x) = 1. From this point on we tacitly assume the group axioms. The trivial variety E, determined by the law x = 1, is the isomorphism class of the one element group. All other varieties of groups are nontrivial. Every nontrivial variety V of groups admits, for each cardinal r, groups free of rank r relative to V. We adopt the notation Fr(V) for a fixed, but arbitrary, group free of rank r relative to V. Any two such groups (for fixed V and r) are isomorphic.

Convention: If E is the trivial variety and r is any cardinal, then Fr(E) is the trivial group 1.

If G and H are groups then we say that G universally covers H provided every universal sentence of Lo true in G is also true in H. Note that H :::; G is a sufficient condition for G to universally cover H. We say that G and H are universally equivalent or have the same universal theory and write G ="1 H provided G universally covers Hand H universally covers G. An existential sentence of Lo is one of the form :lXip(x) where x is a tuple of distinct variables and ip(x) is a formula of Lo containing no quantifiers and containing free at most the variables in x. A primitive sentence of Lo is an existential sentence of Lo equivalent modulo the group axioms to one of the form :lx(l\j(Uj(x) = 1) 1\ I\k(Vk(X) -=I- 1)) where x is a tuple of distinct variables and the Uj(x) and Vk(X) are words in at most the variables in x. Since the negation of a universal sentence of Lo is equivalent to an existential sentence of Lo and vice-versa G ="1 H may be paraphrased as asserting that every universal sentence and every existential sentence of Lo true in G is also true in H (and vice-versa). It is easy to see that if G1 ="1 G ="1 G2 , then G 1 :::; H :::; G2 is a sufficient condition for G ="1 H. A somewhat more sophisticated sufficient condition is the following. H :::; G and every finite system

Uj(X) = 1,1:::;j:::; J

Vk(X) -=I- 1,1 :::; k :::; K

of equations and inequations ( in finitely many variables x = (x 1, ... , X n ) )

which has a solution in G must also have a solution in H. To see that this is so observe that G universally covers Hi so, it will suffice to show that every existential sentence of Lo true in G must also be true in H. Now we may assume that the matrix ip(x) of the existential sentence :lXip(x) is written in disjunctive normal form modulo the group axioms Vi(l\j(Ui,j(X) = 1) 1\ I\k(Vi,k(X) -=I- 1)). The sentence is then equivalent to

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the disjunction VaX(!\j(Ui,j(X) = 1) !\ !\k(Vi,k(X) =I- 1)) of primitive sen­tences. Since a disjunction is true provided at least one of the disjuncts is, it suffices to prove that every primitive sentence of Lo true in G is also true in H; hence, the sufficiency of the above criterion is established.

We say that two groups G and H are elementarily equivalent and write G = H provided G and H satisfy the same sentences of Lo. A theorem of Vaught (Theorem 4 of Chapter 6, Section 38 in [G]) asserts that, if V is any variety and rand s are infinite cardinals, then Fr(V) = Fs(V). In particular, Fr(V) ='<1 Fs(V) when rand s are infinite.

Let n be a positive integer. The Burnside variety Bn of exponent n is the variety of groups determined by the law xn = 1. Adian proved that, for all sufficiently large odd n,

(B1) Fr(Bn) is infinite for all r ~ 2; moreover, every finite subgroup of Fr(Bn) is cyclic.

and (B2) The centralizer of every nontrivial element in Fr(Bn) is cyclic for

all r ~ 2. Sirvanjan proved that, for all sufficiently large odd n, (B3) Fw(Bn) embeds in F2(Bn). We shall call an integer n > 0 an Adian-Sirvanjan integer provided

(B1), (B2) and (B3) hold for Bn. A prime Adian-Sirvanjan integer p shall be an Adian-Sirvanjan prime.

Every member G of a variety V of groups is a homomorphic image of a group Fr(V) free in V. If there is an epimorphism 'ljJ : Fr(V) ~ G such that r is finite and K ere 'ljJ) is the normal closure in Fr (V) of finitely many elements of Fr(V), then G is finitely presented relative to V.

3. Varieties and Discrimination

We begin by remarking that, although we chose to live in the world of groups, the results of this section go through in the context of universal algebra.

Let V be a nontrivial variety of groups. Let Fw (V) be a group free of count ably infinite rank relative to V. Suppose {aI, a2, ... } = {an+l : n < w} freely generates Fw(V) relative to V.

Definition 3.1. [NJ A group G E V discriminates V provided G dis­

criminates Fw(V).

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Now G E V discriminates V just in case, given finitely many elements

wk(al, ... , an) =1= 1

in Fw(V), there is a homomorphism 'ljJ : Fw(V) --+ G such that 'ljJ( Wk( aI, ... , an)) =1= 1 for all k. This is equivalent to the following. Given finitely many words Wk(Xl, ... , xn) such that none of the equations

Wk(XI, ... ,Xn ) = 1

is a law in V, there is a tuple (g1, ... , gn) E Gn such that Wk(gl, ... , gn) =1= 1 for all k.

Convention: The trivial group 1 discriminates the trivial variety E.

Definition 3.2. Let V be a variety of groups. V is finitely discrim­inable provided there is a finitely generated group G E V such that G discriminates V.

Lemma 3.1. V is finitely discriminable if and only if there is a positive

integer r such that Fr (V) discriminates V.

Proof: If G = Fr(V) discriminates V for some integer r > 0, then V is discriminated by an r-generator member; hence, it is finitely discriminable. Suppose V is finitely discriminable. Suppose r is a positive integer and G = (b l , ... , br) E V discriminates V. Let Fr(V) be freely generated relative to V by aI, ... , ar· Then we get an epimorphism 'ljJ : Fr(V) --+ G, ai 1-+ bi, 1 :::; i :::; r.

Suppose that Wk(Xl, ... , xn) are finitely many words such that none of the equations Wk(XI, ... , xn) = 1 is a law in V. Then there are elements gi = ui(b1, ... , br ), 1 :::; i :::; n in G such that Wk(UI(bl , ... , br ), ... , un(b1 , ... , br )) =1= 1 for all k. It follows that Wk( UI (aI, ... , ar ), ... , Un (aI, ... , ar )) =1= 1 in Fr(V) for all k. Hence, Fr(V) discriminates V .•

Definition 3.3. Let V be a finitely discriminable variety of groups. Then min{l :::; r < w : Fr(V) discriminates V} is the index of discrimination of V. If m is the index of discrimination of V, then D(V) = {Fr(V) : m :::; r < w}.

Theorem 3.1. fGS} Let V be a finitely discriminable variety of groups.

Let r ;::: 1 be a cardinal. Then Fr(V) ='1 Fs(V) for all cardinals s ;::: r if and only if Fr (V) discriminates V. In particular, if m is the index of

discrimination of V, then Fm(V) ='1 Fs(V) for all m:::; s:::; w.

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Theorem 3.2. Let V be a finitely discriminable variety of groups with

index of discrimination m. Let G E V be Fm(V) inclusive. If D(V) dis­criminates G, then G ='1 Fm(V).

Proof: Assume D(V) discriminates G. It will suffice to show that, if

Uj(XI, ... ,Xn ) = 1

Vk(XI, ... , Xn) -I- 1

has a solution (gl, ... ,gn) E Gn, then it has a solution over Fm(V). Since D(V) discriminates G there is an integer r ~ m and a homomorphism 'lj; : G ----) Fr(V) such that 'lj;(Vk(gl, ... ,gn)) -I- 1 for all 1 ::; k ::; K. It follows that the primitive sentence 3XI, ... ,xn(Aj(uj(XI, ... ,Xn) = 1) A

Ad Vk (Xl, ... , Xn) -I- 1)) is true in Fr (V). But since Fr (V) ='1 F m (V) the above primitive sentence must also be true in F m (V). Hence, the system has a solution over F m (V) .•

Corollary 3.1. Let V be a finitely discriminable variety of groups with

index of discrimination m. Let G E V be finitely presented relative to V and suppose that G is Fm(V) inclusive. Then G ='1 Fm(V) if and only if

D(V) discriminates G.

Proof: One direction follows immediately from the theorem. Suppose G E

V is finitely presented relative to V, is Fm(V) inclusive and G ='1 Fm(V), Suppose < al, ... ,an;RI, ... ,RJ >vis a finite presentation of G relative to V. Let wk(al, ... , an), 1 ::; k ::; K be finitely many nontrivial ele­ments of G. Then the primitive sentence 3XI, ... ,xn(Aj(Rj(XI, ... ,Xn) = 1) A Ak (Wk (Xl, ... , Xn) -I- 1)) holds in G; hence, it holds in F m (V) and there is (bl , ... , bn) E Fm(v)n such that

Rj(b l , ... , bn ) = 1 1::; j ::; J

wk(h, ... , bn ) -I- 1 1::; k ::; K.

It follows that the assignment ai I---> bi , 1 ::; i ::; n extends to a homo­morphism'lj; : G ----) Fm(V) such that 'lj;(wk(al, ... , an)) -I- 1, 1 ::; k ::; K.

• 4. The Variety 0 of All Groups

In this section we merely review known results about universally free groups (see Definition 4.1). These results will be contrasted later with results for the groups G ='1 F2(Bp) where p is an Adian-Sirvanjan prime. If 0 is the variety of all groups and r ~ 1 is a cardinal, then we write Fr for

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Fr(O). Fw embeds in F2. For example, the commutator subgroup [F2' F2] of F2 is free of countably infinite rank. Now let 2 S; r S; w. Then Fw ~ [F2' F2] S; Fr S; Fw from which it follows that Fr ='1 Fs for all cardinals 2 S; r < s. Thus 2 is the index of discrimination of O. The universal equivalence of the nonabelian free groups suggests the possibility of their elementary equivalence. Of course their universal equivalence is a far cry from their elementary equivalence.

Suppose R is a commutative ring with 1. Within the category of unital R-modules an object P is projective just in case every short exact sequence

O-+N-+M-+P-+O

splits. This is easily seen to be equivalent to P being a direct summand in a free R-module. Now suppose V is a variety of groups. We define a group P E V to be projective relative to V provided every short exact sequence of groups in V,

1 -+ K -+ G -+ P -+ 1,

splits. Essentially the same proof shows that this is equivalent to P being a retract of a group free in V. By the Neilsen-Schreier subgroup theorem, a group P is projective relative to the variety of all groups if and only if it is free. The Neilsen-Schreier subgroup theorem also implies that a group is freely separated (discriminated) if and only if it is residually (fully residually) free. Suppose G is a nonabelian residually free group. Suppose gh =f=. hg in G. Then their commutator [g, h] = g-lh-1gh is nontrivial. Thus, there is a free group Fr and an epimorphism 'ljJ : G -+ Fr such that ['ljJ(g) , 'ljJ(h)] = 'ljJ([g, h]) =f=. 1. Hence, Fr is nonabelian and r ~ 2. Since Fr is projective it is a retract in G and F2 S; Fr S; G; so, G is F2-inclusive. Nonabelian free groups, of course, satisfy the existential sentence ::Ix, y( xy =f=. yx). Other properties of nonabelian free groups are that they are CT and even CSA. Here a group is GT or commutative transitive provided the centralizers of nontrivial elements coincide with the maximal abelian subgroups. That is rendered by the universal sentence

\:Ix, y, z(((y =f=. 1) 1\ (xy = yx) 1\ (yz = zy)) -+ (xz = zx)).

Moreover, a group is GSA or conjugately separated abelian provided maximal abelian subgroups are malnormal. That is equivalent to being CT and satisfying the universal sentence

\:Ix,y,z(((x =f=.1) 1\ (xy = yx) 1\ (Z-lyzx = xz-1yz)) -+ (xz = zx)).

Free groups are CSA.

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Lemma 4.1. fE] A GT residually free group is GSA.

Proo!' Suppose G is eT and residually free. Let a E G\ {I} and suppose b,g-lbg E Gc(a) = {x E G : ax = xa}. Suppose to deduce a contra­diction that 9 rf- Gc(a). Then [g,a] =I=- 1. Thus there is a free group F and an epimorphism 'l/J : G ----+ F such that ['l/J(g), 'l/J(a)] = 'l/J([g, a]) =I=- 1. But this is impossible. From ['l/J(g), 'l/J(a)] =I=- 1 we have 'l/J(a) =I=- 1. Moreover 'l/J(b), 'l/J(g)-l'l/J(b)'l/J(g) E GF('l/J(a)) and F is eSA. Hence, 'l/J(g) commutes with 'l/J(a) - a contradiction. Hence g E Gc(a) and Gis eSA .•

Theorem 4.1. fE] Let G be a nonabelian residually free group. The fol­lowing three conditions are equivalent in pairs.

(1) G is fully residually free. (2) G is GT. (3) G is GSA.

Proo!' Lemma 4.1 has already established the equivalence of (2) and (3). It will suffice to show that (1) and (2) are equivalent. Suppose G is fully residually free. Let b E G\{l} and suppose a, c E Gc(b). Assume to deduce a contradiction that ac =I=- ca Then [a, c] =I=- 1. Thus there is a free group F and an epimorphism 'l/J : G ----+ F such that 'l/J(b) =I=- 1 and ['l/J(a), 'l/J(c)] = 'l/J([a, cD =I=- 1. But this is impossible. 'l/J(a)'l/J(b) = 'l/J(b)'l/J(a), 'l/J(b)'l/J(c) = 'l/J(c)'l/J(b) and F is CT; hence, 'l/J(a)'l/J(c) = 'l/J(c)'l/J(a) - a contra­diction. Therefore ac = ca and G is eT.

Now suppose G is CT. The proof will proceed by induction on the cardinality n of S = {gl, ... , gn} <;;; G\ {I} no element of which is annihilated in a free homomorphic image. The result is true when n = 1 since G is residually free. Now suppose n > 1 and the result is true for all 1 :::; k < n. Suppose first that S is not contained in an abelian subgroup of G. Then some pair of elements of S, which we may take to be gn-l and gn, does not commute. Thus T = {gl, ... ,gn-2,[gn-l,gn]} is contained in G\{l} and, by inductive hypothesis, there is a free group F and an epimorphism 'l/J : G ----+ F such that 'l/J does not annihilate any element of T. But then 'l/J cannot annihilate any element of S either. It remains to treat the case where the gi commute in pairs ,which hypothesis we now assume. Since G is a residually free CT group it is eSA by Lemma 4.1. We claim that there is some 9 E G such that g-lgng does not commute with gn-l. Otherwise, since G is eSA, gn-l would be central in G. But a nonabelian eT group must be centerless; so, we have arrived at a contradiction. The claim is

140

established. Pick one such g. Hence U = {gl, ... ,gn-2,[gn-1,g-lgng]} is contained in G\{l}. By inductive hypothesis there is a free group F and an epimorphism 'ljJ : G -4 F such that 'ljJ does not annihilate any element of U. But then 'ljJ cannot annihilate any element of S either. That completes the induction .•

Definition 4.1. A group G ='<1 F2 is universally free.

Theorem 4.2. (R) A finitely generated group is universally free if and

only if it is nonabelian and fully residually free.

Theorem 4.3. (KM1) A finitely generated fully residually free group is

finitely presented.

Definition 4.2. Let G be a CT group and let a E G\{l}. Let M = Cc(a). Then the HNN-extension

(G, t; rel(G), C 1mt = m \1m E M)

is a free rank 1 centralizer extension of G.

The above construction preserves fully residually freeness. In particular, we have

Theorem 4.4. If Go is fully residually free then so is every free rank 1

centralizer extension of Go.

Fro of: By Definition 4.2, we take b E G o\{l} and let M = Cco(b). So that our free rank 1 centralizer extension of Go is the group G,

G = (Go, t; rel(Go ), C 1zt = z \lz EM).

Start off by viewing G as the amalgamated free product

G = Go *M (M x (t; ).

We need to show that G is fully residually free. For that purpose let gl, ... , gk

be finitely many nontrivial elements of G. Using the normal form for free products with amalgamation (see [MKSj), we may write for each j = 1, ... , k

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where N(j) :?: 0, bi,j E Go \M, mi,j E iZ\{O}, and Zj E M. Note that bi,j E Go \M is equivalent to [bi,j, b] =f. 1. Now since Go is fully residually free, there is a free group F and an epimorphism 'P : Go ----) F such that ['P(bi,j),'P(b)] = 'P([bi,j,b]) =f. 1 for all i,j. This forces 'P(bi,j) =f. 1 and 'P(b) =f. 1.

Let GF('P(b)) = (u). Suppose that 'P(Zj) = uej for all j, 1:::; j :::; k. Now for each positive integer n E N, we may define an extension 'ljJn : G ----) F of 'P by 'ljJn \co= 'P, 'ljJn(t) = un.

Now fix a j, 1 :::; j :::; k. Could we have 'ljJn(gj) = 1 for infinitely many n E N? Suppose to deduce a contradiction that there were infinitely many n E N such that 'ljJn(gj) = 1. Then

'P(bo,j )uml,jn'P(b1,j )"''P(bN (j)-l,j )umN(i),jn+ej = 1

for infinitely many values of n. But then by G. Baumslag's "Big Powers Lemma" (see Proposition 1 [GB]), we then conclude that

for some i, with 0 :::; i :::; N(j) - 1. Thus for that 'P(bi,j) we must have that 'P(bi,j) E GF(u) = GF('P(b)) and so ['P(bi,j),'P(b)] = 1. This contradicts our choice of 'P.

The above contradiction shows that the set

is a cofinite subset of N (Le., its complement S; = N\Sj is finite). Since this is so for all j, 1 :::; j :::; k, we must have the finite intersection

Sl n ... n Sk =f. ¢.

(Note if Sl n··· n Sk were empty, then (Sl n··· n Sk)' = S~ u··· u S~ = N - which is impossible since S~ U ... U S~ is a finite union of finite sets.)

Choose n E Sl n ... n Sk. Then 'ljJn(gj) =f. 1 for all j with 1 :::; j :::; k. Hence G is fully residually free .•

Theorem 4.5. [KM1] A finitely generated group G is fully residually free

if and only if there is a finite rank free group Go and a finite sequence Go :::; G 1 :::; ... :::; Gn of free rank 1 centralizer extensions such that G is isomorphic to a finitely generated subgroup of Gn .

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5. The Burnside Varieties

Let n be an Adian-Sirvanjan integer. Then Fw(Bn) embeds in F2(Bn). So, if 2 ::; r ::; w, then F2(Bn) embeds in Fr(Bn) which, in turn, embeds in F2(Bn). It follows that Fr(Bn) =\;/ F2(Bn) for all 2 ::; r ::; w. Thus 2 is the index of discrimination of Bn. Moreover, since the centralizer of every nontrivial element in Fr(Bn) is cyclic, the relatively free groups Fr(Bn) are CT. Now suppose that p is an Adian-Sirvanjan prime.

Lemma 5.1. Suppose G E Bp and G is Gp x Gp exclusive. If G is GT, then G is GSA.

Proof: Suppose G E B p, G is GpxGp exclusive and G is CT. Let a E G\{l}. Then Gc(a) = (a) ~ Gp. Suppose g-Ibg E (a) for some 1 i- b E (a). Since (a) = (b) we may assume b = a. Then g-Iag = am and a = g-PagP = amP. But we may compute exponents modulo p and, by Fermat's Little Theorem, amP = am. Therefore, am = a and g-Iag = a. So 9 E Gc(a) = (a). Hence, Gis CSA .•

Corollary 5.1. The relatively free groups Fr(Bp) are GSA.

Proof: Since the centralizer of every nontrivial element is isomorphic to Gp, one has that Fr(Bp) is CT and Gp x Gp exclusive .•

More generally, if n is any Adian-Sirvanjan integer, then the free groups Fr(Bn) are CSA. To see that let G = Fr(Bn) where r 2': 2. Let a E G\ {I}. Let Gc(a) = (b). Suppose g-I(b)g intersects (b) nontrivially. Since G is CT we must have, in that event, g-I(b)g = (b). Consider the subgroup H = (g, b) ::; G. Observe (b) is normal in H and the quotient H / (b) is generated by the image of 9 - an element of finite order. Then IHI = I (b) I [H : (b) 1 < 00.

Then H must be cyclic. But Gc(a) = (b) ::; H is maximal cyclic. Hence, H = (b) and 9 E (b) = Gc(a). It follows that Gc(a) is malnormal in G. Hence, G is CSA.

Theorem 5.1. Suppose G E Bp is GT and Gp x Gp exclusive. If G is separated by D(Bp), then G is discriminated by D(Bp).

Proof: The proof will be by induction on the cardinality n of S = {gl, ... ,gn} <;;; G\{l} no element of which is annihilated by a homomor­phism into a group free in Bp' We have the result for n = 1 since G is separated by D(Bp ). Now suppose n > 1 and the result is true for all k with 1 ::; k < n. Assume first that S is contained in an abelian subgroup

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of G. Then (gl) = ... = (gn) ~ Cp and so gi = g'{'i where p does not divide mi for 2 ::; i ::; n. Now, since D(Bp) separates G, there is r ::::: 2

and a homomorphism '¢ : G -; Fr(Bp) such that ,¢(gl) # 1. But then ,¢(gi) = ,¢(gl )mi # 1 for all 2 ::; i ::; n. Hence, '¢ does not annihilate any element of S. Now suppose at least one pair of elements of S does

not commute. We may assume that gn-1 and gn do not commute. Then

T = {gl, ···,gn-2, [gn-1,gn]} is contained in G\{l} By inductive hypothesis there is r ::::: 2 and a homomorphism '¢ : G -; Fr (Bp) such that '¢ does annihilate any element of T. But then '¢ does not annihilate any element of S either. That completes the induction .•

Observe that, since F2(Bp) satisfies the universal sentence I::/x, y(((x "l­I) 1\ (xy = yx)) -; V~:~(y = xk)), every group G ='1 F2(Bp) is Cp x Cp

exclusive. More generally, if n is any Adian-Sirvanjan integer and G ='1

F2 (Bn ), then, for every 9 E G\{l}, one has that Ca(g) is cyclic. To see that consider the universal sentences

(ul) I::/X1,X2((X1X2 = X2X1) -; VO::;k 1 ,k2,ml,m2<n((X1 = (X'{'lX~2)kl) 1\

(X2 = (X'{'lX~2)k2)) and

(u2) I::/x1, ... , Xn+1(l\i<j(XiXj = XjXi) -; Vi<j(Xi = Xj)).

Both hold in F2 (Bn ) since every abelian subgroup of F2 (B n ) is cyclic of order at most n. Hence they both hold in G. (ul) asserts that every abelian subgroup is locally cyclic and (u2) asserts that every abelian subgroup has at most n elements.

Suppose that n is a composite Adian-Sirvanjan integer. Let 1 < d < n be a divisor of n. Let B be freely generated in Bn by {a1, a2, a3} and let A be the subgroup generated (necessarily freely) by {a1, a2}. Let G be the

subgroup of B generated by {a1,a2,al}. SinceF2 (Bn ) = A ::; G ::; B =

F3(Bn) ='1 F2(Bn) we have G ='1 F2(Bn). But the finitely generated group G cannot be free in Bn since its abelianization, isomorphic to Cn x Cn X Cd, has order n 2d which is not a power of n. For an Adian-Sirvanjan prime p

are there any finitely generated G ='1 F2 (Bp) which are not free in Bp? We ponder that question in the next section.

We conclude this section with a proof that cyclicity of finite subgroups

is inherited by models of the universal and existential theory of the free Burnside groups.

Theorem 5.2. Let n be an Adian-Sirvanjan integer and assume G ='1

F2(Bn). Then every finite subgroup of G is cyclic.

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Proof' For each positive ineger N the universal sentence

\Ix 1 , ... , xN(Ai$.j Vk (XiXj = Xk) -; Ai<j(XiXj = XjXi))

holds in F2(Bn) since every finite subgroup is abelian. Thus every finite subgroup of G is abelian. But we have already seen that every abelian subgroup of such G is cyclic .•

6. A Possible Non-Free Model and a Question of Philip Hall

If G is a group and H ::; G let HG be the normal closure of H in G. If H, K ::; G let [H, K] be the subgroup generated by {[h, k] : (h, k) E Hx K}.

If G 1 and G2 are groups let G 1 * G2 be their free product. Let V be a variety of groups and G be a group. Let V(G) be the intersection of the family of subgroups K normal in G such that GIK E V. Then V(G),

the verbal subgroup of G corresponding to V, is fully invariant in G and is the least normal subgroup K in G such that G IKE V. We define (following Hanna Neumann [N] and Magnus, Karrass, Solitar [MKS]) the verbal product G 1 *v G2 as r/([G1, G 2]r n Vcr)) where r = G 1 * G2.

If G 1, G2 E V then G 1 *v G2 E V and each of G 1 and G 2 embeds in G 1 *VG2' Moreover *v restricted to groups in V is the coproduct in V. That means essentially that if G 1 , G 2, G E V then every pair of homomorphisms 1/;i : G i -; G, i = 1,2, uniquely determines a homomorphism 1/; : G 1 *VG2 -; G.

Among other results one has that Fr(V) is the verbal product relative to V of r copies of F 1(V). Moreover, if G 1 , G 2 E V, then each of G 1 and G 2 is a retract of G 1 *v G2. We observe G 1 is a retract since if 1/; : G 1 *v G2 -; G 1 is the homomorphism determined by 1/;1 : G 1 -; G 1 and 1/;2 : G2 -; G 1 where 1/;1 is the identity automorphism and 1/;2 is the trivial map 1/;2 (x) = 1 for all x, then 1/; is a retraction from G 1 *v G2 onto G 1 . Similarly G 2 is a retract of G 1 *v G 2. Furthermore, G 1 *v G 2 is generated by the embedded images of G 1 and G 2 and, if Hi ::; Gi , i = 1,2, then the subgroup (H1' H2) of G1 *v G2 has the verbal product decomposition H1 *v H2.

Now let p be an Adian-Sirvanjan prime. Let *pdenote the verbal prod­uct with respect to the variety Bp and let Kp be the Kostrikin variety of locally finite groups satisfying the law xP = 1. Let Kp be the verbal subgroup operator corresponding to Kp- Suppose r ~ 2 is finite. Then Fr(Kp) = Fr(Bp)/Kp(Fr(Bp)) is a finite group. Since Kp(Fr(Bp)) has fi­nite index in the finitely generated group Fr(Bp) it must itself be finitely

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generated. However, I\:p(Fr(Bp)) is perfect (i.e. it coincides with its com­mutator subgroup). Therefore it cannot be free in Bp. This is so since the abelianization of Fd(Bp) for any cardinal d ?: 1 is a vector space of dimen­sion d over the p element field. Now let F = F4(Bp) be freely generated relative to Bp by a1, a2, a3 and a4. Let A be the subgroup generated (neces­sarily freely) bya3 and a4 and let B be the subgroup generated (necessarily freely) by a1 and a2. Let C = I\:p(A) so that C is finitely generated and perfect. F = B *p A. Consider the subgroup 0 = (B, C) = B *p C. Now F2(Bp) = B :::; 0 :::; F = F4(Bp) ='V F2(Bp). It follows that the finitely generated group 0 is universally equivalent to F2(Bp). Observe that, if 0 were free in B p, then the retract C of 0 would be projective relative to Bp.

Here we observe a connection with Problem 21 of [NJ, which problem is attributed to Philip Hall. The question posed is the following. Suppose V is a variety of groups of exponent zero or prime power. If P E V is projective relative to V must P be free in V? Note that a positive answer to Philip Hall's question in the case of exponent an Adian-Sirvanjan prime would imply that 0 is not free in Bp.

Kovacs and Newman [KN] report that Philip Hall's question has a neg­ative answer in the case of exponent zero; however, to the best of our knowledge the question remains open for prime power exponent. Other conditions would also imply that 0 is not free in Bp. Suppose we de­fine the Rank of a group to be the minimum cardinality of a set of gen­erators. A consequence of the Grushko-Neumann Theorem asserts that Rank(Ol * O2) = Rank(Ol) + Rank(02)' Now C = C' :::; 0' so 0/0' ~ Cp x Cpo If 0 were free it would have rank 2 and, since it is nonabelian, Rank( 0) = 2 also under the assumption of freeness. But, if it were the case that Rank(Ol *p O2) = Rank(Ol) + Rank(02) for all 0 1, O2 E B p, we would have Rank(O) = Rank(B) + Rank(C) = 2 + Rank(C) and Rank( C) > 0 since C -=J- 1. Thus, if the analog of the Grushko-Neumann corollary holds for B p, then 0 cannot be free in Bp. Suppose the finite rank free groups Fr(Bp), 2:::; r < ware Hopfian. As just argued, if 0 were free in B p , then rank(O) = 2. Say 0 were freely generated by bland b2. Now let 'Ij; : 0 -+ 0 be the endomorphism determined by 'lj;1 : B -+ 0, 'lj;2 : C -+ 0 where 'lj;1 is the homomorphism determined by ai f---4 bi , i = 1,2, and 'lj;2 is the trivial map 'lj;2(X) = 1 for all x. Then 'Ij; is an epi-endomorphism with 1 -=J- C :::; Ker('Ij;). That would contradict the Hopf property. Hence, if the free groups Fr(Bp), 2 :::; r < ware Hopfian, then 0 cannot be free in Bp. As far as we know it also is an open question as to whether or not these free groups Fr(Bp) are Hopfian.

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7. Questions

Let G be a group and let n be a positive integer, Let (Xl, .. " Xn; ) be free on the n distinct elements Xl, .. " Xn , Let wE G* (Xl, .. " Xn; ), View the formal expression w = 1 as an equation over G in the variables Xl, .. " X n , Now every assignment Xi I---? gi E G, i = 1, .. " n, extends to a unique retraction 'ljJ : G * (Xl, .. " Xn; ) .-, G, Call the tuple (gl, .. " gn) E Gn a solution to w = 1 provided w E Ker('ljJ) , For each subset S ~ G * (Xl, .. " Xn; ), let Vc(S) ~ Gn be the solution set to the system w = 1, w E S of equations, G is equationally Noetherian provided for every positive integer n and every subset S ~ G * (Xl, .. " Xn; ) there is a finite subset So ~ S such that Vc(S) = Vc(So),

Question 1: If n is an Adian-Sirvanjan integer and r 2: 2 is an integer must Fr(Bn) be equationally Noetherian?

Question 2(Philip Hall): If V is a variety of groups of prime power exponent and P E V is projective relative to V must P be free in V?

Question 3: Suppose we define the Rank of a group to be the minimum cardinality of a set of generators, Let p be an Adian-Sirvanjan prime and let *p be the coproduct in B p' If G l , G2 E Bp must Rank( Gl *p G 2 )

Rank(Gl ) + Rank(G2 )?

Question 4: If p is an Adian-Sirvanjan prime and r 2: 2 is an integer must Fr(Bp) be Hopfian?

Question 5: If n is an Adian-Sirvanjan integer and H E Bn is finitley generated and universally equivalent to F2 (Bn ) must H be embeddable in some Fr(Bn)?

Question 6: If n is an Adian-Sirvanjan integer and 2 ~ r < s ~ w

must Fr(Bn) == Fs(Bn)?

8. References

[A] S,l. Adian," Classification of periodic words and their application in group theory," Springer Lecture Notes in Mathematics 806, Burnside Groups, J,L, Mennicke, Editor, Springer-Verlag, Berlin (1980), 1 -40,

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[B] B. Baumslag,"Residually free groups," Proc. London Math. Soc. (3) 17 (1967), 402 - 418.

[GB] G. Baumslag, "On generalised free products," Math. Z. 78 (1962), 423-438.

[BMR] G.Baumslag, A.G. Myasnikov and V.N. Remeslennikov, "Alge­braic geometry over groups I. Algebraic sets and ideal theory," J. Alg. 219 (1999), 16 - 79.

[G] G. Gratzer, Universal Algebra, Van Nostrand, Princeton (1968).

[GS] A.M. Gaglione and D. Spellman,"The persistence of universal for­mulae in free algebras," Bull. Austral. Math. Soc. 36 (1987), 11 - 17.

[K] A.I. Kostrikin,"The Burnside problem," Izv. akad. Nauk. Ser. Mat. 23 (1959), 3 - 34.

[KM1] O. Kharlampovich and A.G. Myasnikov,"Irreducible affine vari­eties over a free group: II Systems in quasi-quadratic triangular form and description of residually free groups," J. Alg. 200 (1998), 517 - 570.

[KM2] O. Kharlampovich and A.G. Myasnikov,"Tarski's problem about the elementary theory of free groups has a positive solution," Electron. Res. Announc. Amer. Math. Soc. 4 (December 1998), 101 - 108.

[KN] L.G. Kovacs and M.F. Newman,"Hanna Neumann's problems on varieties of groups," Springer Lecture Notes in Mathematics 372, Proc. In­temat. Conf. Theory of Groups, Canberra (1973), 417 - 431.

[MKS] W.Magnus, A. Karrass and D. Solitar, Combinatorial Group The­ory, Dover, Mineola (2004).

[N] H. Neumann, Varieties of Groups, Springer - Verlag, New York (1967).

[Ne] P.M. Neumann," Splitting groups and projectives in varieties of groups," QJ Maths (Oxford) 18 (1967), 325 - 352.

[R] V.N. Remeslennikov,"3-Free groups," Siberian J. Math. 30 (6) (1989), 153 - 157.

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[Se] Z. Sela," Diophantine geometry over groups VI: The elementary the­ory of a free group," GAFA, in press.

lSi] V.L. Sirvanjan,"Imbedding of the group B(oo,n) in the group B(2, n)," Math. USSR-Jzv. 4 (1977), 181 - 199.

Changing generators in non free groups

Richard Goldstein

Department of Mathematics, The University of Albany, Albany, NY 12222. USA E-mail: richardg@albany.edu

Let Fn be a free group of rank n with basis X = {Xl, X2,'" ,xn }. If Y = {Yl, Y2, ... ,Yn} is any other set of generators for Fn then they form a basis for Fn since finitely generated free groups are Hopfian. In this paper which shall show that this fact characterizes finitely generated free groups.

Keywords: free group, generators, cut vertex

Introduction Let Fn be a free group of rank n and X ={Xl, X2,'" ,xn } be a basis for

Fn. Let {Ul,U2,'" ,un} be a set of n freely reduced words in X. A stan­dard fact is that {Ul' U2,'" ,un} is another basis for Fn if and only if the endomorphism <I> where <I>(Xi) = Ui for i = 1,2" " ,n is an isomorphism. One property of fg free groups is that they are Hopfian. This asserts that any endomorphism which is surjective is an isomorphism. Thus, we can conclude that {Ul,U2,'" ,un} is a basis for Fn if and only if they generate

Fn. Finitely generated free groups can be characterized in many distinct

ways, for example they freely act on trees, they are the fundamental group of a finite graph, etc. The purpose of this paper is to give another char­acterization, or more precisely to characterize what it means for a finitely generated group G to not be free.

Main Theorem Let W be a freely reduced word in X, we say that w is primitive if we

can find words {W2,'" ,wn } such that {W,W2,'" ,wn } is a basis for Fn. A classical construction associates to a freely reduced word W in X a

finite graph, called the coinitial (star) graph of w, see for example. l If r is a finite graph and v is a vertex of r then v is said to be a cut vertex of r if its removal disconnects r. Whiteheads" Cut vertex theorem" ,2 asserts that if W is primitive, then the coinitial graph of w has a cut vertex. Moreover if

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the coinitial graph of w has a cut vertex and w is not a single generator or its inverse, then there exists an automorphism ¢ of Fn such that the length of ¢(w) is less than that of w. Thus by iteration if w is not primitive then there exists an automorphism 'IjJ of Fn such that 'IjJ(w) has no cut vertex.

If v is a cyclically reduced word which starts with Xi to some positive power and the coinitial graph of v has no cut vertex then the word XiV also has no cut vertex since its coinitial graph contains the coinitial graph of v and thus XiV is non-primitive. It is essential for the coinitial graph of a non-primitive word v to have no cut vertex to guarantee that XiV is non-primitive. For example abab is not primitive in the free group F2 with basis {a, b} but a2bab is primitive since {a2bab, ab} generates F2 and is therefore a basis.

Let G be a group of rank n which is not free. Let < Xl, X2, ... ,Xn Irl' r2, ... > be a presentation of G. Suppose that one of the relators say rl is a primitive element in Fn with basis {Xl, X2, ... ,xn }. Let ¢ be an automorphism of Fn such that ¢(rl) = Xl. Now < Xl, X2,'" ,xnlxl' ¢(r2), ¢(r3),'" > is another presentation of G. We can now eliminate Xl from the generating set and therefore G has rank less than n. Hence we may assume that in any presentation of G with n generators then no relator is primitive.

Let ¢ be an automorphism of Fn such that ¢(rl) is cyclically re­duced and has no cut vertex. Now G has a presentation of the form < Xl, X2,'" ,xnl¢(rl), ¢(r2),'" >. Without loss of generality let us assume that ¢(rl) starts with Xl to some positive power. Clearly {xI¢(rl),x2,'" ,xn } generates G and thus we have the following result.

Theorem 0.1. Let G be a group of rank n which is not free and let X = {XI,X2,'" ,xn } be a set of generators of G. There exists a new set

of generators for G, {UI,U2,'" ,un}, where Uj is a word in X such that

{UI' U2,'" , un} is not a basis for Fn , the free group with basis X. In partic­ular they do not generate Fn. Moreover we can make UI be a non-primitive word in Fn.

References

1. Lyndon, R., Schupp, P. "Combinatorial Group Theory." Springer-Verlag, (1977).

2. Whitehead, J., H., C. "On certain sets of elements in a free group." Proc. London Math. Soc. 41, (1936), 48-56.

Matrix Completions over Principal Ideal Rings

William H. Gustafson

Texas Tech University, Lubbock, Texas

Donald W. Robinson

Brigham Young University, Provo, Utah

R. Bruce Richter

University of Waterloo, Waterloo, Ontario

William P. Wardlaw

U. S. Naval Academy, Annapolis, Maryland

Dedicated to Anthony M. Gaglione on his sixtieth birthday and to the memory of William H. Gustafson

Abstract

We show that if A is a k x n matrix over a principal ideal ring R, with k < n, and if d is any element of the ideal generated by the k x k minors of A, then A forms the top k rows of an n x n matrix of determinant d. This parallels a 1981 result of Gustafson, Moore, and Reiner, and continues a program initiated by Hermite in 1849. Then we use these results to obtain an extension of a 1997 result of Richter and Wardlaw for good matrices.

1. Introduction

If A is a k x n matrix with k < n, the matrix completion problem intiated

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by Hermite asks if A can be completed to an n x n matrix with prescribed determinant d. Gustafson, Moore, and Reiner, at the beginning of [5], give a brief summary of the history of the problem of completing a k x n matrix with k < n over certain commutative rings to an n x n matrix over the same ring with appropriate determinant. They also include references to some of the principal players in this program initiated by Hermite in 1849. In our contribution below, we show in Theorem 1 that principal ideal rings are among the rings over which this matrix completion is always possible. Theorem 2 states the relationships between six properties of a k x n matrix over a commutative ring. It extends a similar theorem in [9] by giving a best possible exposition of these relationships. Finally we show in Theorem 3 that if such completions are always possible over each ring in a given collection of rings, then they are also always possible over the unrestricted direct product of the collection.

Throughout this paper, R will denote a commutative ring with identity. If A is a k x n matrix over R with k :S n, then Dk(A) denotes the ideal of R generated by the k x k sub determinants of A. We say that A has left block form if A is equivalent over R to a matrix E = [L 0], where L is a k x (k + 1) matrix over Rand 0 is the k x (n - k - 1) zero matrix over R. That is, there are matrices P E GL(k, R) and Q E GL(n, R) such that PAQ = [L 0]. Note that if k = n - 1 or k = n, the 0 block is missing and E = L; indeed, we can take A = E = L.

2. Results

The following lemma was proved but not explicitly stated in [5], and was used to prove their main result. For the sake of completeness, we include a proof here.

Lemma. Let A be a k x n matrix over the commutative ring R with identity, let k < n, and let d E Dk(A). If A has left block form over R, then A enlarges to an n x n matrix A * over R whose determinant is d and whose top k rows form the matrix A.

Proof. Let P E GL(k, R) and Q E GL(n, R) be such that PAQ = E = [L 0], where L is k x (k + 1). Clearly, Dk(A) = Dk(E) = Dk(L). Let Cj = (_l)k+l+ j det(Lj ), where Lj is the k x k submatrix of L obtained by deleting the jth column of L. Thus we can write d E Dk(A) as a linear combination d = L. ajcj = det(L *), where L * is the (k+ 1) x (k + 1) matrix

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obtained from L by adding [al a2 ... ak+l] as its last row. Let p = det(P) and q = det(Q), and multiply the last row of L* by the unit pq to obtain the matrix M* with det(M*) = pdq. Now let E* be the direct sum of M* and the (n - k - 1) x (n - k - 1) identity matrix In-k-l; thus,

E* = [~* In_Ok-J = [;]

is an n x n matrix over R with det(E*) = pdq whose first k rows form the matrix E = P AQ. It follows that the matrix

*=[P-l ° ] * _l=[P-IEQ-1]=[A] A ° In- k- l E Q FQ-l A'

has det(A*) = d and its first k rows form the matrix A. D

Our first main result is

Theorem 1. Suppose that R is either a Dedekind domain or a principal ideal ring, and that A is a k x n matrix over R with k < n. If d is any element of Dk(A), then there is an nxn matrix A* over R with determinant det(A*) = d whose first k rows form the matrix A.

Proof. In view of our lemma, we need only establish that A has a left block form over R for each of the two cases.

When R is a Dedekind domain, Theorem 1 is the main result of [5], where they proved a lemma that every k x n matrix over a Dedekind do­main R has a left block form over R. They comment that this lemma was established in a more general form by Levy [6] in 1972.

When R is a principal ideal ring, W. C. Brown shows in [2, Thm. 15.24, p. 194], that every matrix over R has a Smith normal form. When A is k x n over R with k < n, its Smith normal form is a left block form for A ~R. D

Since every principal ideal domain is also a Dedekind domain, Theorem 1 only extends the result of [5] when R is a principal ideal ring with nonzero divisors of zero.

We were especially interested in the connection between Theorem 1 and the 1997 result [9] regarding good matrices. In [9], R was a commutative ring with identity and an r x n matrix A over R was defined to be left good if, for every vector x in RlXT, the ideal (xA) generated by the entries in the vector xA is the same as the ideal (x) generated by the entries of the

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vector x. Our lemma allows us to extend the Main Theorem of [9] to our second main result.

Theorem 2. Consider the following statements about an r x n matrix A over the commutative ring R with identity.

(1) The rows of A extend to a basis of RIxn.

(2) A can be enlarged to a matrix A* E GL(n, R). (3) A has a Smith normal form [Ir 0]. (4) A has a right inverse over R. (5) Dr(A) = R. (6) A is left good.

Then

(a) The statements (1), (2), and (3) are equivalent over any commutative ring R with identity.

(b) The statements (4), (5), and (6) are equivalent over any commutative ring R with identity.

(c) The statement (3) implies the statement (4) but in general they are not equivalent.

(d) If A has left block form then all six statements are equivalent.

Proof. Theorem 2 (a), (b), and (c) was proved in [9], except for the implications (2) =} (3) and (5) =} (4), and the fact that (4) ~ (3).

The statement (2) means that there is an (n - r) x n matrix A' over R and an n x n matrix B* over R such that

A*= [~,] and A * B* = I is the n x n identity matrix. But then it is clear that AB* = [Ir 0] is a Smith normal form for A. That is, (2) =} (3).

The implication (5) =} (4) is immediate from [8, Cor. 1.28, p. 84]. How­ever, for the sake of completeness we give the following elementary proof. If M is any m x n matrix over R and v = (CI, ... , cr ) a vector of column indices of M, so that 1 :::; Cj :::; n, we let M(v) denote the m x r subma­trix of M whose jth column is the cjth column of M. It is easy to see that if In is the n x n identity matrix, then M(v) = M In(v). Now each r-subset {CI' ... ,cr } of {I, 2, .. , ,n} with 1 :::; CI < C2 < ... < Cr :::; n corresponds uniquely to a vector v = (CI, ... , Cr), and we can number these vectors (perhaps lexicographically) VI, V2, ... , VN with N = (~).

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Let dj = det(A j ) with Aj = A(vj). Then Dr(A) = R implies 1 = 'L. bjdj for scalars b1, b2 , ... , bN in R. Now, for each j = 1, 2, ... , N, let B j be the n x r matrix B j = In(vj)Adj(Aj ), and let B be the n x r matrix B = 'L.bjBj. Then

AB = I)jABj = LbjAIn(Vj)Adj(Aj)

= L bjAjAdj(Aj ) = L bjdjIr = Ir

and (4) A has a right inverse B. That is, (5) =} (4). The following example from [4], attributed to Kaplansky in [1, p. 7],

shows that (4) =f? (3) in Theorem 2 (c), and hence the result of Theorem 2 (c) is the best possible. Let R be the ring of polynomials in x, y, z over the real numbers modulo the ideal generated by x 2 + y2 + z2 - 1. This is the ring of polynomial functions on the standard 2-sphere in 3-space. The 1 x 3 matrix A = [x y z] has a right inverse AT. If it had a Smith normal form [1 0 0], then there would be a matrix Q E GL(3, R) such that AQ = [1 0 0]. Assume such a Q exists with last column q = [f 9 hjT. Then A q = xf +yg+zh = 0 for all points on the 2-Sphere. Thus q provides a tangent vector field to the 2-sphere which, because of independence of the columns of Q, is never zero on the 2-sphere. But no such vector field exists, as is shown in [3, p. 70]. This contradiction shows (4) =f? (3). (In fact, the same argument shows directly that A does not have a left block form, since that would require an invertible Q with AQ = [u v 0].) This completes the proof of Theorem 2 (a), (b), and (c).

To establish (d), we first observe that when r = n, the implication (4) =} (2) is a tautology. Then we use our lemma to show that (5) =}

(2) when r < n and A has a left block form over R. It is clear from (5) that 1 E R = Dr(A). By our lemma, A can be enlarged to a matrix A* with determinant 1 when r < n. It is well known that a matrix over a commutative ring R with identity is invertible over R if and only if its determinant is a unit in R. (See [7, Thm. 50, p. 158].) Hence (5) =} (2). 0

We remark that if A is an (n - 1) x n matrix over R, then it is already in left block form, so statements (1) - (6) of Theorem 2 are equivalent. In particular, if A has a right inverse, then it extends to an n x n matrix which is invertible over R. (The latter was shown using an outer product argument in [9].)

Recall that in the proof of Theorem 1 we observed that if R was either a principal ideal ring or a Dedekind domain, then every r x n matrix over

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R with r :::::: n had a left block form. Thus we have the following corollary to Theorem 2.

Corollary. If R is a principal ideal ring or a Dedekind domain, then statements (1) - (6) of Theorem 2 are equivalent.

This corollary extends the Main Theorem of [9] from principal ideal rings to rings which are either principal ideal rings or are Dedekind domains. Our next theorem allows further extension of the class of rings for which certain properties mentioned above hold.

Let R be a commutative ring with identity. Then R has property L if every r x n matrix A over R with r :::::: n has a left block form over R. R has property C if every r x n matrix A over R with r < n has, for each dE Dr(A) an n x n completion A* with det(A*) = d. R has property G if statements (1) - (6) of Theorem 2 are equivalent for every r x n matrix A over R with r :::::: n. Note that L =} C =} G, by our Lemma and Theorem 2.

Theorem 3. Let P be anyone of the properties L, C, or G, and let R =

EBjRj be the unrestricted direct sum of the commutative rings R j (j E J), where each R j has identity 1 j. Then R has property P if and only if each R j has property P.

Proof. We consider R to be an internal direct sum, so each Rj is a subring and an ideal of R. For each a E R, aj = alj denotes the projection of a into Rj; we call aj the j-component of a. Thus (aj h = 0 if j =f. k and (aj)j = aj for all j, k E J. If A is a matrix over R, then we let Aj = ljA be the matrix of the same size over R j obtained by replacing each entry in A by its j-component. We write Ai to denote a matrix chosen with entries in R j , to distinguish it from the j-component Aj = ljA obtained from a matrix A already chosen with entries in R. In the proofs below, we will often define a matrix A over R by first specifying a matrix Aj over R j

for each j E J, and letting A be the matrix of the same size over R with j-component Aj = ljA = Ai.

Now suppose that R has property L and that Aj E (Rjyxn with r :::::: n. Since Aj E Rrxn, there are matrices P E GL(r, R) and Q E GL(n, R) such that PAjQ = E = [L 0], with L E w x (r+l). But Ai = ljA' implies that PAjQ = P(ljA')Q = (ljP)(Aj)(ljQ) = PjAjQj = E = E j = [L j 0] with Pj E GL(r, R j ) and Qj E GL(n, R j ). Thus, R j has property L.

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On the other hand, suppose that for each j E J, R j has property L, and that A E Rrxn with r s: n. Then Aj E (Rjyxn for each j E J,

and so there are matrices Pj E GL(r, Rj ) and Qj E GL(n, Rj) such that PjAjQj = [Lj Ol with Lj E (Rjyx(r+l). Let P E wxr be the matrix with j-component IjP = Pj and let Q E Rnxn be the matrix with j_ component IjQ = Qj for every j E J. It is easy to see that P E GL(r, R), Q E GL(n, R), and PAQ = [L Ol with L E Rrx(r+l) such that IjL = Lj for each j E J. Thus, R has property L.

Now suppose that R has property C and that Aj E (Rj)rxn with r < n and d E Dr(A~). Since Aj E Rrxn, there is an A* E Rnxn whose first R rows form the matrix Aj and with determinant det(A*) = d. But the first R rows of IjA* = (A*)j E (Rj)nxn also form the matrix Aj and det((A*)j) = d = dj . Thus, Rj has property C.

On the other hand, suppose that for each j E J, Rj has property C, A E Rrxn with r < n, and d E Dr(A). For each j E J, IjA = Aj E (Rjyxn has ljd = dj E Dr(Aj) and has an n x n completion (Aj)* over Rj with det((Aj)*) = d = dj . Let A* be the n x n matrix over R with IjA* =

(A*)j = (Aj)* for each j E J. Since det((A*)j) = dj for each j E J, it follows that det(A*) = d. Since the first r rows of (A*)j form the matrix Aj for each j E J, it follows that the first R rows of A * form the matrix A. That is, A* is the n x n completion of A with determinant d. Hence, R has property C.

Suppose R has property G and that Aj, (Bj)T E (Rjyxn satisfy AjBj = (Ir)j, which is statement (4) of Theorem 3 for the ring Rj . Let

E = [Ir Ol- [Ir Olj, A = E+Aj, and B = ET +Bj. Note that A = [Ir 0li if i -=1= j, Aj = Aj, and similarly for B. Then AB = Ir shows that A satisfies ( 4) for the ring R. Since R has property G, A must also satisfy (2), so A has an invertible completion A* over R. It follows that (A*)j E GL(n,Rj ) is the n x n completion of Aj = Aj over Rj . Thus, (4) =} (2) in Rj , so Rj has property G.

Finally, suppose for each j E J that R j has property G and that A, BT E Rrxn satisfy AB = I r . Then AjBj = (Ir)j for each j E J, and so property G ensures that each Aj can be completed to an (Aj)* E GL(n, Rj ). Now let A* be the n x n matrix over R with j-component IjA* = (A*)j =

(Aj )*. Then A* E GL(n,R) and its first R rows form the matrix A. Thus (4) =} (2) in R, so R has property G. 0

158

References

1. H. Bass, Introduction to some methods of algebraic K-theory, CBMS 20, Amer. Math. Soc., Providence, RI, 1974.

2. W. C. Brown, Matrices over Commutative Rings, Dekker, New York, 1992. 3. M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, New York,

1967. 4. W. H. Gustafson, P. R. Halmos, and J. M. Zelmanowitz, The Serre Conjecture,

Amer. Math. Monthly 85 (1978), 357-359. 5. W. H. Gustafson, M. E. Moore, and I. Reiner, Matrix completions over

Dedekind rings, Linear and Multilinear Algebra 10 (1981), 141-144. 6. L. S. Levy, Almost diagonal matrices over Dedekind domains, Math. Z. 124

(1972), 89-99. 7. N. H. McCoy, Rings and Ideals, Mathematical Association of America, Wash­

ington, 1965. 8. B. R. McDonald, Linear Algebra over Commutative Rings, Dekker, New York,

1984. 9. R. B. Richter and W. P. Wardlaw, Good matrices: matrices which preserve

ideals, Amer. Math. Monthly 104 (1997) , 932-938.

A primer on computational group homology and cohomology using GAP and SAGE

David Joyner

Department of Mathematics, US Naval Academy, Annapolis, MD, wdj@usna.edu.

Dedicated to my friend and colleague Tony Gaglione on the occasion of his sixtieth birthday

These are expanded lecture notes of a series of expository talks survey­ing basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what Z[G] is (where G is a group written multi­plicatively and Z denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference or proof of everything.

Several computer algebra packages are used to illustrate the computa­tions, though for various reasons we have focused on the free, open source packages, such as GAP [Gap] and SAGE [St] (which includes GAP). In par­ticular, Graham Ellis generously allowed extensive use of his HAP [EI] documentation (which is sometimes copied almost verbatim) in the pre­sentation below. Some interesting work not included in this (incomplete) survey is (for example) that of Marcus Bishop [Bi], Jon Carlson [C] (in MAGMA), David Green [Gr] (in C), Pierre Guillot [Gu] (in GAP, C++, and SAGE), and Marc Roder [Ro].

Though Graham Ellis' HAP package (and Marc Roder's add-on HAPcryst [RoJ) can compute comhomology and homology of some infinite groups, the computational examples given below are for finite groups only.

1. Introduction

First, some words of motivation.

159

160

Let G be a group and A a G-modulea .

Let AC denote the largest submodule of A on which G acts trivially. Let us begin by asking ourselves the following natural question.

Question: Suppose A is a submodule of a G-module B and x is an arbitrary G-fixed element of BfA. Is there an element bin B, also fixed by G, which maps onto x under the quotient map?

The answer to this question can be formulated in terms of group co­homology. ("Yes", if Hl(G, A) = 0.) The details, given below, will help motivate the introduction of group cohomology.

Let Ac is the largest quotient module of A on which G acts trivially. Next, we ask ourselves the following analogous question.

Question: Suppose A is a submodule of a G-module Band b is an arbitrary element of Bc which maps to 0 under the natural map Bc ---+

(B f A)c. Is there an element a in ac which maps onto b under the inclusion map?

The answer to this question can be formulated in terms of group homol­ogy. ("Yes", if H1(G, A) = 0.) The details, given below, will help motivate the introduction of group homology.

Group cohomology arises as the right higher derived functor for A t--------+

A c. The cohomology groups of G with coefficients in A are defined by

(See §4 below for more details.) These groups were first introduced in 1943 by S. Eilenberg and S. MacLane [EM]. The functor A t--------+ AC on the category of left G-modules is additive and left exact. This implies that if

is an exact sequence of G-modules then we have a long exact sequence of cohomology

0---+ AC---+Bc ---+ CC ---+ Hl(G,A)---+ Hl(G,B) ---+ Hl(G,C) ---+ H2(G,A) ---+ ...

(1)

aWe call an abelian group A (written additively) which is a left Z[G]-module a G­module.

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Similarly, group homology arises as the left higher derived functor for

A f------+ Ae. The homology groups of G with coefficients in A are defined by

Hn(G,A) = Tor~[el(Z,A).

(See §5 below for more details.) The functor A f------+ Ae on the category of left G-modules is additive and right exact. This implies that if

is an exact sequence of G-modules then we have a long exact sequence of homology

..• ----+ H2(G,C) ----+ H1(G,A) ----+ H1(G,B)----+ Hl(G,C) ----+ Ae ----+ Be ----+ Ce ----+ o. (2)

Here we will define both cohomology Hn(G, A) and homology Hn(G, A) using projective resolutions and the higher derived functors Ext n and Tor n.

We "compute" these when G is a finite cyclic group. We also give various functorial properties, such as corestriction, inflation, restriction, and trans­fer. Since some of these cohomology groups can be computed with the help of computer algebra systems, we also include some discussion of how to

use computers to compute them. We include several applications to group

theory. One can also define Hl(G, A), H2(G, A), ... , by explicitly construct­

ing co cycles and coboundaries. Similarly, one can also define HdG,A), H 2 (G,A), ... , by explicitly constructing cycles and boundaries. For the proof that these constructions yield the same groups, see Rotman [R], chap­

ter 10. For the general outline, we follow §7 in chapter 10 of [R] on homology.

For some details, we follow Brown [B], Serre [S] or Weiss [W]. For a recent expository account of this topic, see for example Adem [A].

Another good reference is Brown [B].

2. Differential groups

In this section cohomology and homology are viewed in the same frame­work. This "differential groups" idea was introduced by Cartan and Eilen­berg [CE], chapter IV, and developed in R. Godement [G], chapter 1, §2. However, we shall follow Weiss [W], chapter 1.

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2.1. Definitions

A differential group is a pair (L, d), L an abelian group and d : L -t L a homomorphism such that d2 = O. We call d a differential operator. The group

H(L) = Kernel (d)jlmage (d)

is the derived group of (L, d). If

then we call L graded. Suppose d (more precisely, diLJ satisfies, in addi­tion, for some fixed r -I- 0,

We say d is compatible with the grading provided r = ±l. In this case, we call (L, d, r) a graded differential group. As we shall see, the case r = 1 corresponds to cohomology and the the case r = -1 corresponds to homology. Indeed, if r = 1 then we call (L, d, r) a (differential) group of cohomology type and if r = -1 then we call (L, d, r) a group of homology type. Note that if L = EB~=_ooLn is a group of cohomology type then L' = EB~_ooL~ is a group of homology type, where L~ = L-n' for all n E Z.

For the impatient: For cohomology, we shall eventually take L = EBnHomc(Xn, A), where the Xn form a chain complex (with +1 grad­ing) determined by a certain type of resolution. The group H(L) is an abbreviation for EBnExt Z[c] (Z, A). For homology, we shall eventually take L = EBnZ@Z[C] X n , where the Xn form a chain complex (with -1 grading) determined by a certain type of resolution. The group H(L) is an abbrevi­ation for EBn Tor ~[C] (Z, A).

Let (L,d) = (L,dL) and (M,d) = (M,dM) be differential groups (to be more precise, we should use different symbols for the differential operators of Land M but, for notational simplicity, we use the same symbol and hope the context removes any ambiguity). A homomorphism f : L -t M satisfying do f = f 0 d will be called admissible. For any nEZ, we define nf : L -t M by (nf)(x) = n· f(x) = f(x) + .,. + f(x) (n times). If f

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is admissible then so is nf, for any n E Z. An admissible map f gives rise to a map of derived groups: define the map f* : H(L) ~ H(M), by f*(x + dL) = f(x) + dM, for all x E L.

2.2. Properties

Let f be an admissible map as above.

(1) The map f* : H(L) ~ H(M) is a homomorphism. (2) If f : L ~ M and 9 : L ~ M are admissible, then so is f + 9 and we

have (J + g)* = f* + g*. (3) If f : L ~ M and 9 : M ~ N are admissible then so is go f : L ~ N

and we have (g 0 J)* = g* 0 f*. (4) If

(3)

is an exact sequence of differential groups with admissible maps i, j then there is a homomorphism d* : H(N) ~ H(L) for which the following triangle is exact:

H(L)

/ H(N)

(4)

H(M) This diagramb encodes both the long exact sequence of cohomology (1) and the long exact sequence of homology (2). Here is the construction of d*: Recall H (N) = Kernel (d) jlmage (d), so any x E H (N) is represented by an n E N with dn = O. Since j is surjective, there is an m E M

bThis is a special case of TMoreme 2.1.1 in [G].

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such that j(m) = n. Since j is admissible and the sequence is exact, j(dm) = d(j(m» = dn = 0, so dm E Kernel(j) = Image (i). Therefore, there is an £ E L such that dm = i(£). Define d*(x) to be the class of £ in H(L), i.e., d*(x) = £ + dL. Here's the verification that d* is well-defined: We must show that if we defined instead d* (x) = £' + dL, some £' E L, then £' - £ E dL. Pull back the above n E N with dn = 0 to an m E M such that j (m) = n. As above, there is an £ E L such that dm = i(£). Represent x E H(N) by an n' E N, so x = n' + dN and dn' = O. Pull back this n' to an m' E M such that j(m') = n'. As above, there is an £' E L such that dm' = i(£'). We know n' - n E dN, so n' - n = dn", some n" E N. Let j(m") = n", some m" E M, so j(m'-m-dm") = n' = n-j(dm") = n'-n-dj(m") = n'-n-dn" = O. Since the sequence L - M - N is exact, this implies there is an £0 E L such that i(£o) = m' - m - dm". But r~(£o) = i(d£o) = dm' - dm = ief') - i(£) = i(£' - f), so f' - £ E dL.

(5) If M = L ffi N then H(M) = H(L) ffi H(N).

(6)

proof: To avoid ambiguity, for the moment, let dx denote the differen­tial operator on X, where X E {L,M,N}. In the notation of (3), j is projection and i is inclusion. Since both are admissible, we know that dMIL = dL and dMIN = dN. Note that H(X) C X, for any differential group X, so H(M) = H(M) n L ffi H(M) nNe H(L) ffi H(N). It follows from this that that d* = O. From the exactness of the triangle (4), it therefore follows that this inclusion is an equality. o Let L, L', M, M', N, N' be differential groups. If

o -----. L ~M ~N --0

fl 91 hI (5)

o -----. L' ., ~M'

., -.L- N' --0

is a commutative diagram of exact sequences with i, i', j, j', j, g, h all admissible then

H(L)

1·1 H(L')

~ H(M)

9·1 ., ~ H(M')

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commutes,

H(M) j. H(N) ---.

9·1 h·l ./

H(M') '. H(N') ---. commutes, and

H(N) d. H(L) ---.

h·l 1·1 H(N') d. H(L') ---.

commutes. This is a case of Theorem 1.1.3 in [W] and of Theoreme 2.1.1 in [G]. The proofs that the first two squares commute are similar, so we only verify one and leave the other to the reader. By assumption, (5) com­mutes and all the maps are admissible. Representing x E H(M) by x = m + dM, we have

h*j*(x) = h*(j(m) + dN) = hj(m) + dN' = gi'(m) + dN'

= g*(i'(m) + dM') = g*i:(m + dM) = g*i:(x),

as desired. The proof that the last square commutes is a little different than this, so we prove this too. Represent x E H(N) by x = n + dN with dn = 0 and recall that d*(x) = £+dL, where dm = i(£), £ E L, where j(m) = n, for m E M. We have

On the other hand,

d*h*(x) = d*(h(n) + dN') = f!' + dL',

for some f!' E L'. Since h(n) EN', by the commutativity of (5) and the definition of d*, £' E L' is an element such that i'(£') = gi(£). Since i' is injective, this condition on £' determines it uniquely mod dL'. By the commutativity of (5), we may take f!' = J(£).

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(7) Let L, L', M, M', N, N' be differential graded groups with grading +1 (i.e., of "cohomology type"). Suppose that we have a commutative diagram, with all maps admissible and all rows exact as in (5). Then the following diagram is commutative and has exact rows:

This is Proposition 1.1.4 in [W]. As pointed out there, it is an immediate consequence of the properties, 1-6 above. Compare this with Proposition 10.69 in [R].

(8) Let L, L', M, M', N, N' be differential graded groups with grading -1 (Le., of "homology type"). Suppose that we have a commutative diagram, with all maps admissible and all rows exact, as in (5). Then the following diagram is commutative and has exact rows:

-j_. ~ Hn{N) _d_. ~ H n _ 1 (L)

f. J . __ ~ H n +1 (N') _d_. ~ HnCL') _i~~ Hn(M') _j_~ ~ Hn(N') _d_. ~ Hn_1(L') __ ~ .

This is the analog of the previous property and is proven similarly. Compare this with Proposition 10.58 in [R].

(9) Let (L, d) be a differential graded group with grading T. If dn = dl Ln

then dn +r 0 dn = 0 and

(6)

is exact. (10) If {Ln I nEil} is a sequence of abelian groups with homomorphisms

dn satisfying (6) then (L, d) is a differential group, where L = EBnLn and d = EBndn.

2.3. Homology and cohomology

When T = 1, we call Ln the group of n-cochains, Zn = Ln n Kernel (dn ) the group of n-cocycles, and Bn = Ln n dn-1(Ln- 1) the group of n­

coboundaries. We call Hn(L) = Zn/Bn the nth cohomology group. When T = -1, we call Ln the group of n-chains, Zn = LnnKernel (dn ) the group of n-cycles, and Bn = Ln ndn+1 (Ln+1) the group of n-boundaries. We call Hn(L) = Zn/Bn the nth homology group.

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3. Complexes

We introduce complexes in order to define explicit differential groups which will then be used to construct group (co)homology.

3.1. Definitions

Let R be a non-commutative ring, for example R = Z[G]. We shall define a "finite free, acyclic, augmented chain complex" of left

R-modules. A complex (or chain complex or R-complex with a negative grading)

is a sequence of maps

X 0,,+1 X On 0,,-1 X ... -; n+l -; n ---t X n - 1 -; n-2 -; ... (7)

for which OnOn+l = 0, for all n. If each Xn is a free R-module with a finite basis over R (so is ~ Rk, for some k) then the complex is called finite free. If this sequence is exact then it is called an acyclic complex. The complex is augmented if there is a surjective R-module homomorphism E : Xo -; Z and an injective R-module homomorphism f1. : Z -; X-I such that 00 = f1. 0 E, where (as usual) Z is regarded as a trivial R-module.

The standard diagram for such an R-complex is

82 81 80 8_ 1 . .. -------+ X 2 -------+ X 1 -------+ Xo -------+ X -1 ---+ X - 2 -------+ ...

z ___ Z

1 r o o

Such an acyclic augmented complex can be broken up into the positive part

and the negative part

'71 JJ. X 0-1 X 0_2 X o -; ~ -; -1 -; -2 -; -3 -; ...

Conversely, given a positive part and a negative part, they can be combined into a standard diagram by taking 00 = f1. 0 E.

168

If X is any left R-module, let X* = HomR(X, Z) be the dual R­module, where Z is regarded as a trivial R-module. Associated to any f E HomR(X, Y) is the pull-back f* E HomR(Y*, X*). (If y* E y* then define f* (y*) to be y* 0 f : X ---> Z.) Since "dualizing" reverses the direction of the maps, if you dualize the entire complex with a -1 grading, you will get a complex with a +1 grading. This is the dual complex.

When R = Z[G] then we call a finite free, acyclic, augmented chain complex of left R-modules, a G-resolution. The maps Oi : Xi ---> X i - 1 are sometimes called boundary maps.

Remark 3.1. Using the command BoundaryMap in the GAP CRIME package of Marcus Bishop, one can easily compute the boundary maps of a coho­mology object associated to a G-module. However, G must be a p-group.

Example 3.1. We use the package HAP [El] to illustrate some of these concepts more concretely. Let G be a finite group, whose elements we have ordered in some way: G = {9b ... , 9n}.

Since a G-resolution X* determines a sequence of finitely generated free Z[G]-modules, to concretely describe X* we must be able to con­cretely describe a finite free Z[G]-module. In order to represent a word w in a free Z[G]-module M of rank n, we use a list of integer pairs w = [[i 1,el],[i2,e2], ... ,[ik,ek]]. The integers ij lie in the range {-n, ... ,n} and correspond to the free Z[G]-generators of M and their additive inverses. The integers ej are positive (but not necessarily distinct) and correspond to the group element gej'

Let's begin with a HAP computation.

r---------------------------GAP--------------------______ ~ gap> LoadPackage ("hap") ; true gap> G:~Group ([ (1,2,3), (1,2) 1);; gap> R:~Reso1utionFiniteGroup(G, 4);;

This computes the first 5 terms of a G-resolution (G = 8 3 )

The bounday maps 8i are determined from the boundary component of the GAP record R. This record has (among others) the following components:

• R! .dimension(k) - the Z[G]-rank of the module X k ,

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• R! . boundary(k, j) - the image in Xk-l of the j-th free generator of Xk,

• R! . elts - the elements in G, • R! . group is the group in question.

Here is an illustration:

r---------------------------- GAP ----------------------------~

gap> R! .group; Group([ (1,2), (1,2,3) ])

gap> R! .elts; [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]

gap> R! .dimension(3); 4

gap> R! .boundary(3,1); [ [ 1, 2 ], [ -1, 1 ]

gap> R! .boundary(3,2); [ [ 2, 2 ], [ -2, 4 ]

gap> R! .boundary(3,3); [ [ 3, 4 ], [ 1, 3 ], -3, 1 ],

gap> R! .boundary(3,4); [ [ 2, 5 ], [ -3, 3 ], [ 2, 4 ],

-1, 1 ] ]

-1, 4 ], [ 2, 1 ], [ -3, 1 ] ]

In other words, X3 is rank 4 as a G-module, with generators {iI, 12, 13, f4} say, and

Now, let us create another resolution and compute the equivariant chain map between them. Below is the complete GAP session:

r----------------------------- GAP ------------------------------

gap> G1 :=Group ([ (1,2,3), (1,2) ]); Group([ (1,2,3), (1,2) ]) gap> G2 :=Group ([ (1,2,3), (2,3) ]); Group([ (1,2,3), (2,3) ]) gap> phi: =GroupHomomorphismBylmages (G1, G2, [ (1,2,3) , (1,2) ], [ (1,2,3) , (2,3) ] ) ; [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ] gap> R1:=ResolutionFiniteGroup(G1, 4); Resolution of length 4 in characteristic 0 for Group ([ (1,2), (1,2,3) ])

gap> R2:=ResolutionFiniteGroup(G2, 4); Resolution of length 4 in characteristic 0 for Group ([ (2,3), (1,2,3) ]) .

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gap> ZP_map:=EquivariantChainMap(Rl, R2, phi); Equivariant Chain Map between resolutions of length 4 .

gap> map := TensorWithlntegers( ZP_map); Chain Map between complexes of length 4 .

gap> Hphi := Homology( map, 3); [ fL f2, f3 1 -> [ f2, f2*f3, fl*f2-2 gap> Abelianlnvariants(Image(Hphi»; [ 2, 3 1 gap> gap> GroupHomology(Gl,3); [ 6 1 gap> GroupHomology(G2,3); [ 6 1

In other words, H (1)) is an isomorphism (as it should be, since the homology is independent of the resolution choosen).

3.2. Constructions

Let R = Z[G].

3.2.1. Bar resolution

This section follows §1.3 in [W]. Define a symbol [.] and call it the empty cell. Let Xo = R[.], so Xo is

a finite free (left) R-module whose basis has only 1 element. For n > 0, let g1, ... , gn E G and define an n-cell to be the symbol [g1, ... , gn]. Let

where the sum runs over all ordered n-tuples in Gn. Define the differential operators dn and the augmentation E, as G­

module maps, by

171

f(g[.]) = 1, 9 E G

d1([g]) = g[.]- [.],

d2([gl,g2]) = gl[g2]- [glg2] + [gl],

n-l dn([gl, ... , gn]) = gl [g2, ... , gn] + I: (-I)i[gl, ... , gi-l, gigi+1, gi+2, ... , gn]

i=1

+ (-I)n[gl,'" ,gn-l],

for n ~ 1. Note that the condition f(g[.]) = 1 for all 9 EGis equivalent to saying f([.J) = 1. This is because f is a G-module homomorphism and Z is a trivial G-module, so f(g[.]) = gf([.]) = 9 . 1 = 1, where the (trivial) G-action on Z is denoted by a '.

The Xn are finite free G-modules, with the set of all n-cells serving as a basis.

Proposition 3.1. With these definitions, the sequence

X d2 X d1 X € '71 ... -+ 2 -+ 1 -+ 0 -+ til -+ 0,

is a free G-resolution.

Sometimes this resolution is called the bar resolutionc . There are two other resolutions we shall consider. One is the closely related "homogeneous resolution" and the other is the "normalized bar resolution".

This simple-looking proposition is not so simple to prove. First, we shall show it is a complex, Le., d2 = O. Then, and this is the most non-trivial part of the proof, we show that the sequence is exact.

First, we need some definitions and a lemma. Let f : L -+ M and 9 : L -+ M be +1-graded admissible maps. We

say f is homotopic to 9 if there is a homomorphism D : L -+ M, called a homotopy, such that

• Dn = DILn : Ln -+ M n+1 ,

• f - 9 = Dd + dD.

CThis resolution is not the same as the resolution computed by HAP in Example 3.1. For details on the resolution used by HAP, please see Ellis [E2J.

172

If L = M and the identity map 1 : L -> L is homotopic to the zero map o : L -> L then the homotopy is called a contracting homotopy for L.

Lemma 3.1. If L has a contracting homotopy then H(L) = o.

proof: Represent x E H(L) by I! E L with dl! = O. But I! = 1 (I!) -O(I!) = dD(£) + Dd(£) = dD(I!). Since D : L -> L, this shows I! E dL, so x = 0 in H(L).D

Next, we construct a contracting homotopy for the complex X* in Proposition 3.1 with differential operator d. Actually, we shall temporar­ily let X-I = Z, X-n = 0 and d_n = 0 for n > 1, so that that the complex is infinite in both directions. We must define D : X -> X such that

• D-I = Dlz : Z -> X o, • Dn = Dlxn : Xn -> X n+l , • eD_ I = 1 on Z, • dIDo + D_Ie = 1 on X o, • dn+IDn + Dn-Idn = 1 in Xn, for n 2: 1.

Define

D_ I (1) = [.],

Do(g[.]) = [g],

n> 1,

Dn(g[gl, ... ,gn]) = [g,gl, ... ,gn]'

and extend to a Z-basis linearly. Now we must verify the desired properties.

n>O,

By definition, for m E Z, eD_I(m) = e(m[.]) = me([.]) = m. Therefore, eD_ I is the identity map on Z.

Similarly,

(dIDo + D-Ie)(g[.]) = dl([g]) + D_ I(1)

= g[.]- [.] + D_1(1) = g[.]- [.] + [.] = g[.].

For the last property, we compute

173

dn+lDn(g[gl, ... ,gn]) = dn+l([g,gl,'" ,gn])

= g[gl, ... ,gn]- [ggl,'" ,gn] n-l

+ 2:) _l)i-l [g, gl, ... ,gi-l, gigi+l, gi+2, ... ,gn] i=1

and

Dn- 1dn(g[gl, ... ,gn])

= Dn- 1(gdn([gl, ... ,gn]))

= Dn- 1(ggl[g2, ... ,gn] n-l

+ 2) -l)ig[gl, ... , gi-l, gigi+l, gi+2,···, gn] i=1

+ (_l)ng[gl"'" gn-l])

= [ggl,g2,'" ,gn] n-l

+ 2:)-l)i[g,gl,'" ,gi-l,gigi+l,gi+2,··. ,gn] i=1

+ (_1)n[g,gl,'" ,gn-l].

All the terms but one cancels, verifying that dn+1Dn + Dn- 1dn = 1 in X n , for n ~ 1.

Now we show d2 = O. One verifies d1d2 = 0 directly (which is left to the reader). Multiply dkDk- 1 + Dk-2dk-l = 1 on the right by dk and dk+1Dk + Dk- 1dk = 1 on the left by dk:

dkDk- 1dk + Dk-2dk-ldk = dk = dkdk+1 Dk + dkDk- 1dk·

Cancelling like terms, the induction hypothesis dk-ldk = 0 implies dkdk+l = O. This shows d2 = 0 and hence that the sequence in Propo­sition 3.1 is exact. This completes the proof of Proposition 3.1. 0

The above complex can be "dualized" in the sense of §3.1. This dualized complex is of the form

'7l M X d- 1 X d-2 X o --+ tfJ --+ -1 --+ -2 --+ -3 --+ ..•

The standard G-resolution is obtained by splicing these together.

174

3.2.2. Normalized bar resolution

Define the normalized cells by

[ J*_{[91, ... ,gnJ, if allgi =/:-1,

gl,···,gn - O'f 1 , 1 some gi = .

Let Xo = R[.J and

n 2': 1,

where the sum runs over all ordered n-tuples in Gn. Define the differential operators dn and the augmentation map exactly as for the bar resolution.

Proposition 3.2. With these definitions, the sequence

X d2 X d1 € '77 ••• ----> 2 ----> 1 ----> Xo ----> u... ----> 0,

is a free G -resolution.

Sometimes this resolution is called the normalized bar resolution. proof: See Theorem 10.117 in [RJ. 0

3.2.3. Homogeneous resolution

Let Xo = R, so Xo is a finite free (left) R-module whose basis has only 1 element. For n > 0, let Xn denote the Z-module generated by all (n + 1)­tuples (gO,.'" gn)· Make Xi into a G-module by defining the action by g: Xn ----> Xn by

9 : (gO, ... , gn) I---> (ggo, . .. , ggn), 9 E G.

Define the differential operators an and the augmentation c, as G­module maps, by

c(g) = 1, n-l

an (go, .,. ,gn) = 2:(-I)i(gO,'" , 9i-l,[Ji,gHl, ... ,gn), i=O

for n 2': 1.

Proposition 3.3. With these definitions, the sequence

175

X 02 X 01 € ••• ----t 2 ----t 1 ----t Xo ----t Z ----t 0,

is a G-resolution.

Sometimes this resolution is called the homogeneous resolution. Of the three resolutions presented here, this one is the most straightfor­

ward to deal with.

proof: See Lemma 10.114, Proposition 10.115, and Proposition 10.116 in [R]. 0

4. Definition of Hn(G, A)

For convenience, we briefly recall the definition of Ext n. Let A be a left R-module, where R = Z[G], and let (Xi) be a G-resolution of Z. We define

Ext z[C](Z, A) = Kernel (d~+l)/Image (d~),

where

d~: Hom(Xn_1,A) ----t Hom(Xn,A),

is defined by sending f : X n- 1 ----t A to fdn : Xn ----t A. It is known that this is, up to isomorphism, independent of the resolution choosen. Recall Ext i[c] (Z, A) is the right-derived functors of the right-exact functor A f-----+ AC = Homc(Z, A) from the category of G-modules to the category of abelian groups. We define

(8)

When we wish to emphasize the dependence on the resolution choosen, we write Hn(G,A,X*).

For example, let X* denote the bar resolution in §3.2.1 above. Call Cn = Cn(G, A) = Homc(Xn, A) the group of n-cochains of G in A, zn = zn(G, A) = Cn n Kernel (8) the group of n-cocycles, and Bn = Bn(G,A) = 8(Cn- 1) the group of n-coboundaries. We call Hn(G,A) = zn / B n the nth cohomology group of G in A. This is an abelian group.

We call also define the cohomology group using some other resolution, the normalized bar resolution or the homogeneous resolution for example. If we wish to express the dependence on the resolution X* used, we write

176

Hn(G, A, X*). Later we shall see that, up to isomorphism, this abelian group is independent of the resolution.

The group H 2 ( G, Z) (which is isomorphic to the algebraic dual group of H2(G,C X )) is sometimes called the Schur multiplier of G. Here C denotes the field of complex numbers.

We say that the group G has cohomological dimension n, written cd(G) = n, if Hn+l(H,A) = 0 for all G-modules A and all subgroups H of G, but Hn(H, A) =I- 0 for some such A and H.

Remark 4.1.

• If cd( G) < 00 then G is torsion-freed. • If G is a free abelian group of finite rank then cd(G) = rank(G). • If cd( G) = 1 then G is free. This is a result of Stallings and Swan (see

for example [RJ, page 885).

4.1. Computations

We briefly discuss computer programs which compute cohomology and some examples of known computations.

4.1.1. Computer computations of cohomology

GAP [Gap] can compute some cohomology groupse. All the SAGE commands which compute group homology or cohomology

require that the package HAP be loaded. You can do this on the command line from the main SAGE directory by typingf

sage -i gap_packages-4.4.10_3.spkg

Example 4.1. This example uses SAGE, which wraps several of the HAP functions . .-----______________________ SAGE

I sage: G = AlternatingGroup(5)

dThis follows from the fact that if G is a cyclic group then Hn(G,7l..) i= 0, discussed below. eSee §37.22 of the GAP manual, M. Bishop's package CRIME for cohomology of p-groups, G. Ellis' package HAP for group homology and cohomology of finite or (certain) infinite groups, and M. Roder's HAPCryst package (an add-on to the HAP package). SAGE [Stl computes cohomology via it's GAP interface. fThis is the current package name - change 4.4.10_3 to whatever the latest version is on http://www.sagemath.org/packages/optional/atthetimeyoureadthis.Also.this command assumes you are using SAGE on a machine with an internet connection.

sage: G.cohomology(l,7) Trivial Abelian Group sage: G.cohomology(2,7) Trivial Abelian Group

4.1.2. Examples

Some example computations of a more theoretical nature.

(1) HO(G,A) = AG. This is by definition.

177

(2) Let L/ K denote a Galois extension with finite Galois group G. We have Hl(G,LX) = 1. This is often called Hilbert's Theorem 90. See Theorem 1.5.4 in [W] or Proposition 2 in §X.1 of [S].

(3) Let G be a finite cyclic group and A a trivial torsion-free G-module. Then Hl(G,A) = O. This is a consequence of properties given in the next section.

(4) If G is a finite cyclic group of order m and A is a trivial G-module then

This is a consequence of properties given below. For example, H2(GF(q)X,C) = O.

(5) If IGI = m, rA = 0 and gcd(r, m) = 1, then Hn(G, A) = 0, for all

n~1.

This is Corollary 3.1.7 in [W]. For example, H 1(A5 ,7L/77L) = O.

5. Definition of Hn(G, A)

We say A is projective if the functor B f----' HomG(A, B) (from the cate­gory of G-modules to the category of abelian groups) is exact. Recall, if

P. d2 P d 1 n € '71 Z = ... --> P2 --> 1 --> r-o --> ~ --> 0 (9)

is a projective resolution of 7L then

178

It is known that this is, up to isomorphism, independent of the resolution choosen. Recall Tor ~[c] (Z, A) are the right-derived functors of the right­exact functor A ~ Ac = Z 0z[c] A from the category of G-modules to the category of abelian groups. We define

(10)

When we wish to emphasize the dependence on the resolution, we write Hn(G, A, Pz).

Remark 5.1. If G is a p-group, then using the command ProjectiveResolution in GAP's CRIME package, one can easily compute the minimal projective resolution of a G-module, which can be either trivial or given as a MeatAxeg module.

Since we can identify the functor A ~ Ac with A ~ A0z[c] Z (where Z is considered as a trivial Z[G]-module), the following is another way to formulate this definition.

If Z is considered as a trivial Z[G]-module, then a free Z[G]-resolution of Z is a sequence of Z[G]-module homomorphisms

satisfying:

• (Freeness) Each Mn is a free Z[G]-module. • (Exactness) The image of Mn+1--Mn equals the kernel of Mn--Mn-l

for all n > O. • (Augmentation) The cokernel of Ml--Mo is isomorphic to the trivial

Z[G]-module Z.

The maps Mn --Mn-l are the boundary homomorphisms of the resolution. Setting TMn equal to the abelian group Mn/G obtained from Mn by killing the G-action, we get an induced sequence of abelian group homomorphisms

... --TMn--TMn_l-- ... --TM1--TMo

This sequence will generally not satisfy the above exactness condition, and one defines the integral homology of G to be

gSee for example http://wvw.math.rwth-aachen.de/-MTX/.

Hn(G,Z) = Kernel(TMn~TMn_l)/Image(TMn+l~TMn)

for all n > o.

5.1. Computations

179

We briefly discuss computer programs which compute homology and some examples of known computations.

5.1.1. Computer computations of homology

Example 5.1. GAP will compute the Schur multiplier H2 (G, Z) using the AbelianlnvariantsMultiplier command. To find H2 (A5,Z), where A5 is the alternating group on 5 letters, type .-_____________________________ GAP ____________________________ --,

gap> AS:=AlternatingGroup(S); Alt ( [ 1 .. 5 J ) gap> AbelianlnvariantsMultiplier(AS); [ 2 J

So, H 2 (A5 , q ~ Z/2Z. Here is the same computation in SAGE:

.-__________________________ SAGE

sage: G = AlternatingGroup(S) sage: G.homology(2) Multiplicative Abelian Group isomorphic to C2

Example 5.2. The SAGE command poincare_series returns the Poincare series of G (mod p) (p must be a prime). In other words, if you input a (finite) permutation group G, a prime p, and a positive integer n, poincare_series(G,p,n) returns a quotient of polynomials f(x) = P(x)/Q(x) whose coefficient of xk equals the rank of the vector space Hk(G, ZZ/pZZ) , for all k in the range 1 ::; k ::; n . r-____________________________ SAGE

sage: G = SymmetricGroup(S) sage: G.poincare_series(2,10)

(x'2 + 1)/(x'4 - x'3 - x + 1) sage: G = SymmetricGroup(3) sage: G.poincare_series(2,lO)

180

1/ (-x + 1)

This last one implies

for 1 ::; k ::; 10.

Example 5.3. Here are some more examples using SAGE's interface to HAP: .-__________________________ SAGE

sage: G = SymmetricGroup(S) sage: G.homology(l) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(2) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(3) Multiplicative Abelian Group isomorphic to C2 x C4 sage: G.homology(4) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(S) Multiplicative Abelian Group isomorphic to C2 x C2 sage: G.homology(6) Multiplicative Abelian Group isomorphic to C2 x C2 sage: G.homology(7)

x C3

x C2

Multiplicative Abelian Group isomorphic to C2 x C2 x C4 x C3 x CS

The last one means that

(Z/2Z)2 x (Z/3Z) x (Z/4Z) x (Z/5Z).

r-----________________________ SAGE

sage: G = AlternatingGroup(S) sage: G.homology(l) Trivial Abelian Group sage: G.homology(1,7) Trivial Abelian Group sage: G.homology(2,7) Trivial Abelian Group

5.1.2. Examples

Some example computations of a more theoretical nature.

181

(1) If A is a G-module then Tor~[G](Z,A) = Ho(G,A) = AG ~ AIDA. proof: We need some lemmas.

Let to : Z[G] ----t Z be the augmentation map. This is a ring homo­morphism (but not a G-module homomorphism). Let D = Kernel (to) denote its kernel, the augmentation ideal. This is a G-module.

Lemma 5.1. As an abelian group, D is free abelian generated by G-1 = {g - 1 I 9 E G}.

We write this as D = Z(G - 1). proof of lemma: If d E D then d = L9EG mgg, where mg E Z and LgEG mg = O. Thus, d = L9EG mg(g - 1), so D c Z(G - 1). To show D is free: If L9EG mg(g - 1) = 0 then L9EG mgg - L9EG mg = 0 in Z[G]. But Z[G] is a free abelian group with basis G, so mg = 0 for all 9 E G. 0

Lemma 5.2. Z 0z[G] A = AIDA, where DA is generated by elements of the form ga - a, 9 E G and a E A.

Recall AG denotes the largest quotient of A on which G acts triviallyh. proof of lemma: Consider the G-module map, A ----t Z0z[G]A, given by a f-------> 10a. Since Z0Z[G] A is a trivial G-module, it must factor through A G. The previous lemma implies AG ~ AIDA. (In fact, the quotient map q : A ----t AG satisfies q(ga - a) = 0 for all 9 E G and a E A, so DA C Kernel (q). By maximality of A G, DA = Kernel (q). QED) SO, we have maps A ----t AG ----t Z 0z[G] A. By the definition of tensor products, the map Z x A ----t A G, 1 x a f-------> 1 . aDA, corresponds to a map Z0z[G] A ----t AG for which the composition AG ----t Z0z[G] A ----t AG is the identity. This forces AG ~ Z 0z[G] A. 0

See also # 11 in §6. (2) If G is a finite group then Ho(G, Z) = Z.

This is a special case of the example above (taking A = Z, as a trivial G-module).

(3) H1(G,Z) ~ G/[G,G], where [G,G] is the commutator subgroup of G.

This is Proposition 10.110 in [R], §10.7. proof: First, we claim: DI D2 ~ G/[G, G], where D is as in Lemma 5.l. To prove this, define e : G ----t DI D2 by 9 f-------> (g-1)+D2. Since gh-1-(g-l) - (h-1) = (g-1)(h-1), it follows that e(gh) = e(g)e(h), so e is

hImplicit in the words "largest quotient" is a universal property which we leave to the reader for formulate precisely.

182

a homomorphism. Since D / D2 is abelian and G / [G, Gj is the maximal abelian quotient of G, we must have Kernel (£I) c [G, Gj. Therefore, £I factors through £I': G/[G,Gj-t D/D2, g[G,G] 1-+ (g-l) +D2. Now, we construct an inverse. Define T : D -t G /[G, G] by 9 -11-+ g[G, G]. Since T(g-l+h-l) = g[G, G]·h[G, G] = gh[G,Gj, it is not hard to see that this is a homomorphism. We would be essentially done (with the construction of the inverse of £I', hence the proof of the claim) if we knew D2 C Kernel (T). (The inverse would be the composition of the quotient D/D2 -t D/Kernel(T) with the map induced from T, D/Kernel(T)-t G/[G,G].) This follows from the fact that any x E D2 can be written as x = (2:g mg(g - 1»(2:h m',,(h - 1» = (2: g,h mgm',,(g - 1)(h - 1»,

so T(X) = I1 h(ghg-lh-l)mgm~[G,Gj = [G,G]. QED (claim) g,

Next, we show H1(G,Z) ~ D/D2. From the short exact sequence

o -t D -t Z[Gj -..:. Z -t 0,

we obtain the long exact sequence of homology

... -t H1(G,D) -t Hl(G,Z[G])-t

H1 (G,Z)!-; Ho(G,D)!.. Ho(G,Z[G]) ~ Ho(G,Z) -t O. (11)

Since Z[Gj is a free Z[Gj-module, H1(G, Z[G]) = O. Therefore a is injective. By item # 1 above (i.e., Ho(G,A) ~ A/DA ~ Ae, we have Ho(G,Z) ~ Ze = Z and Ho(G,Z[G]) ~ Z[G]jD ~ Z. By (11), E* is surjective. Combining the last two statements, we find Z/Kernel (E*) ~ Z.This forces E* to be injective. This, and (11), together imply f must be O. Since this forces a to be an isomorphism, we are done. 0

(4) Let G = F/R be a presentation of G, where F is a free group and R is a normal subgroup of relations. Hopf's formula states: H 2 ( G, Z) ~ (F n R)/[F, RJ, where [F, R] is the commutator subgroup of G. See [RJ, §1O.7. The group H2(G,Z) is sometimes called the Schur multiplier of G.

6. Basic properties of Hn(G, A), Hn(G, A)

Let R be a (possibly non-commutative) ring and A be an R-module. We say A is injective if the functor B 1-+ Home(B, A) (from the category of G­modules to the category of abelian groups) is exact. (Recall A is projective if the functor B 1-+ Home(A, B) is exact.) We say A is co-induced if it has the form Homz(R, B) for some abelian group B. We say A is relatively

183

injective if it is a direct factor of a co-induced R-module. We say A is relatively projective if

7r : Z[G]®z A""""", A, x 121 a f------> xa,

maps a direct factor of Z[G]®z A isomorphically onto A. These are the G­modules A which are isomorphic to a direct factor of the induced module Z[G]®zA. When G is finite, the notions of relatively injective and relatively projective coincidei .

(1) The definition of Hn(G,A) does not depend on the G-resolution X* of Z used.

(2) If A is an projective Z[G]-module then Hn(G, A) = 0, for all n 2: l. This follows immediately from the definitions.

(3) If A is an injective Z[G]-module then Hn(G, A) = 0, for all n 2: l. See also [S], §VII.2.

(4) If A is a relatively injective Z[G]-module then Hn(G,A) = 0, for all n2:l. This is Proposition 1 in [S], §VII.2.

(5) If A is a relatively projective Z[G]-module then Hn(G,A) = 0, for all

n2:l. This is Proposition 2 in [S], §VII.4.

(6) If A = A' EB A" then Hn(G,A) = Hn(G,A') EB Hn(G,A"), for all n 2: O. More generally, if I is any indexing family and A = EBiEI Ai then Hn(G,A) = EBiEIHn(G, Ai), for all n 2: O. This follows from Proposition 10.81 in §10.6 of Rotman [R].

(7) If

is an exact sequence of G-modules then we have a long exact sequence of cohomology (1). See [S], §VII.2, and properties of the ext functor [R], §10.6.

(8) A f------> Hn(G, A) is the higher right derived functor associated to A f------>

AC = Homc(A, Z) from the category of G-modules to the category of abelian groups. This is by definition. See [S], §VII.2, or [R], §1O.7.

iThese notions were introduced by Hochschild [Ho].

184

(9) If

is an exact sequence of G-modules then we have a long exact sequence of homology (2). In the case of a finite group, see [S], §VIII.1. In general, see [S], §VII.4, and properties of the Tor functor in [R], §1O.6.

(10) A ~ Hn(G, A) is the higher left derived functor associated to A ~ AG = Z 0z[G] A on the category of G-modules. This is by definition. See [S], §VII.4, or [RJ, §10.7.

(11) If G is a finite cyclic group then

for all n ~ 1.

Ha(G, A) = AG ,

H2n- 1(G, A) = AG jN A,

H2n(G,A) = Kernel (N)jDA,

To prove this, we need a lemma.

Lemma 6.1. Let G = (g) be acyclic group of order k. Let M = 9 - 1 and N = 1 + 9 + g2 + ... + gk-l. Then

N M N M € ..• --+ Z[G] --+ Z[G] --+ Z[G] --+ Z[G] --+ Z[G] --+ Z[G] --+ Z --+ 0,

is a free G-resolution.

proof of lemma: It is clearly free. Since MN = NM = (g - 1)(1 + 9 + g2 + ... + gk-l) = gk _ 1 = 0, it is a complex. It remains to prove exactness. Since Kernel (€) = D = Image (M), by Lemma 5.1, this stage is exact.

To show Kernel (M) = Image (N), let x = L7':~ mj gj E Kernel (M). Since (g - l)x = 0, we must have ma = ml = ... = mk-l. This forces x = maN E Image (N). Thus Kernel (M) C Image (N). Clearly M N = 0 implies Image (N) C Kernel (M), so Kernel (M) = Image (N). To show Kernel (N) = Image (M), let x = L~;:~ mjgJ E Kernel (N).

Since Nx = 0, we have 0 = €(Nx) = €(N)€(x) = k€(x), so L;;:~ mj = o. Observe that

x = ma' 1 + mIg + m2g2 + ... + mk_lgk- l

= (ma - mag) + (ma + ml)g + m2g2 + ... + mk_lgk- l

= (ma - mag) + (ma + ml)g - (ma + ml)g2 +(ma + ml + m2)g2 - (ma + ml + m2)g3 + ... +(ma + .. + mk_l)gk-l - (ma + .. + mk_l)gk.

185

where the last two terms are actually O. This implies x = -M(ma + (ma+mt)g+(ma+ml +m2)g2+ ... +(ma+ .. +mk_t)gk-1 E Image (M). Thus Kernel (N) C Image (M). Clearly N M = 0 implies Image (M) C

Kernel (N), so Kernel (N) = Image (M). This proves exactness at every stage.D Now we can prove the claimed property. By property 1 in §5.1.2, it suffices to assume n > O. Tensor the complex in Lemma 6.1 on the right with A:

... ---> Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A ~ Z[G] @Z[G) A-=' Z @ Z[G]A ---> 0,

where the new maps are distinguished from the old maps by adding an asterisk. By definition, Z[G] ®Z[G] A ~ A, and by property 1 in §5.1.2, Z ®Z[G] A ~ AIDA. The above sequence becomes

This implies, by definition of Tor,

and

Tor~~Gl(Z, A) = Kernel (N*)/Image (M*) = A[N]I DA.

See also [S], §VIII.4.1 and the Corollary in §VIII.4. (12) The group H2(G, A) classifies group extensions of A by G.

This is Theorem 5.1.2 in [W]. See also §10.2 in [R]. (13) If G is a finite group of order m = IGI then mHn(G, A) = 0, for all

This is Proposition 10.119 in [R]. (14) If G is a finite group and A is a finitely-generated G-module then

Hn( G, A) is finite, for all n 2: 1. This is Proposition 3.1.9 in [W] and Corollary 10.120 in [R].

186

(15) The group Hl(G, A) constructed using resolutions is the same as the group constructed using 1-cocycles. The group H2(G, A) constructed using resolutions is the same as the group constructed using 2-cocycles. This is Corollary 10.118 in [R].

(16) If G is a finite cyclic group then

HO(G,A) = AC,

H 2n- 1(G,A) = Kernel NIDA,

H2n(G,A) = AC INA,

for all n :::: 1. Here N : A --+ A is the norm map N a = L9EC ga and DA is the augmentation ideal defined above (generated by elements of the form ga - a). proof: The case n = 0: By definition, HO(G,A) = Ext~[Cl(Z,A) = Homc(Z,A). Define T: Homc(Z,A) --+ AC by sending f f-----+ f(l). It is easy to see that this is well-defined and, in fact, injective. For each a E AC, define f = fa E Homc(Z,A) by f(m) = mao This shows T is surjective as well, so case n = 0 is proven. Case n > 0: Applying the functor Homc(*,A) to the G-resolution in Lemma 6.1 to get

... <- HomcCZ[G], A) ~ HomcCZ[G], A) ~ HomcCZ[G], A) ~ Homc(Z, A) <- O.

It is known that Homc(Z[G], A) ~ A (see Proposition 8.85 on page 583 of [R]). It follows that

... ~ A ~ A ~ A ~ AC ~ O.

By definition of Ext, for n > 0 we have

and

Ext ifGjl(Z, A) = Kernel (N*)/Image (M*) = Kernel (N)/(g - l)A,

where 9 is a generator of G as in Lemma 6.1. D See also [S], §VIII.4.1 and the Corollary in §VIII.4.

187

(17) If G is a finite cyclic group of order m and A is a trivial G-module then

for all n 2: l.

HO(G,A) = AC ,

H 2n- 1 (G,A) ~ A[m],

H2n(G,A) ~ A/rnA,

This is a consequence of the previous property.

7. Functorial properties

In this section, we investigate some of the ways in which Hn(G, A) depends on G.

One way to construct all these in a common framework is to introduce the notion of a "homomorphism of pairs". Let G, H be groups. Let A be a G-module and Ban H-module. If 0: : H ---+ G is a homomorphism of groups and (3 : A ---+ B is a homomorphism of H-modules (using 0: to regard B as an H-module) then we call (0:,(3) a homomorphism of pairs, written

(0:, (3) : (G, A) ---+ (H, B).

Let G c H be groups and A an H-module (so, by restriction, a G­

module). We say a map

is transitive if fC2,c3fc 1,c2 = fC 1,c2' for all subgroups G1 C G2 C G3 ·

Let X* be a G-resolution and X~ a H-resolution, each with a -1 grading. Associated to a homomorphism of groups 0: : H ---+ G is a sequence of H­

homomorphisms

(12)

Theorem 7.1.

(1) If (0:,(3) : (G,A) ---+ (G',A') and (0:',(3') : (G',A') ---+ (G",A") are homomorphisms of pairs then so is (0:' 00:, (3' 0 (3) : (G, A) ---+ (Gil, A").

188

(2) Suppose (a,(J) : (G,A) ---+ (G', A') is homomorphism of pairs, X* is a G-resolution, and X~ is a G'-resolution (each infinite in both direc­tions, with a -1 grading). Let Hn(G, A, X*) denote the derived groups associated to the differential groups Homc(X*, A) with +1 grading. There is a homomorphism

(a,f3)x.,X; : Hn(G,A,X*) ---+ Hn(G',A',X~)

satisfying the following properties.

(a) IfG = G', A = A', X = X', a = 1 and 13 = 1 then (1, l)x.,x; = 1. (b) If (ex', 13') : (G', A') ---+ (Gil, A") is homomorphism of pairs, X~ is

a Gil -resolution then

(a' 0 a, 13' of3)x.,x;1 = (a', f3')x;,x;1 0 (a, f3)x.,x;.

(c) If (a,'Y) : (G,A) ---+ (G',A') is homomorphism of pairs then

(a,f3 + 'Y)x.,x; = (a,f3)X.,x; + (a, 'Y)x.,x;.

Remark 7.1. For an analogous result for homology, see §§III.8 in Brown [B].

proof: We sketch the proof, following Weiss, [W], Theorem 2.1.8, pp 52-53.

(1): This is "obvious". (2): Let (a,f3) : (G,A) ---+ (G', A') be a homomorphism of pairs. Using

(12), we have an associated chain map

a*: Homc(X*, A) ---+ Homcl(X~,A')

of differential groups (Brown §III.8 in [BD. The homomorphism of coho­mology groups induced by a* is denoted

a~,x.,x; : Hn(G, A, X*) ---+ Hn(G', A', X~).

Properties (a)-(c) follow from §2.2 and the corresponding properties of a*. o

As the cohomology groups are independent of the resolution used, the map (a,f3)x.,x; : Hn(G,A,X*) ---+ Hn(G',A',X~) is sometimes simply denoted by

(13)

189

7.1. Restriction

Let X* = X*(G) denote the bar resolution. If H is a subgroup of G then the cycles on G, en (G, A) =

HomG(Xn(G), A), can be restricted to H: en(H, A) = HomH(Xn(H), A). The restriction map en(G, A) ---7 en(H, A) leads to a map of cohomology classes:

In this case, the homomorphism of pairs is given by the inclusion map 0: : H ---7 G and the identity map f3 : A ---7 A. The map Res is the induced map defined by (13). By the properties of this induced map, we see that Res H,G is transitive: if G c G' C Gil thenj

Res G',G 0 Res Gil ,G' = Res G",G.

A particularly nice feature of the restriction map is the following fact.

Theorem 7.2. If G is a finite group and Gp is a p-8ylow subgroup and if Hn(G, A)p is the p-primary component of Hn(G, A) then

(a) there is a canonical isomorphism Hn(G,A) ~ ffipHn(G,A)p, and (b) Res: Hn(G,A) ---7 Hn(Gp,A) restricted to Hn(G,A)p (identified

with a subgroup of Hn(G, A) via (a)) is injective.

proof: See Weiss, [WJ, Theorem 3.1.15. 0

Example 7.1. Homology is a functor. That is, for any n > 0 and group homomorphism f : G ---7 G' there is an induced homomorphism Hn(f) Hn(G,Z) ---7 Hn(G',Z) satisfying

• Hn(gf) = Hn(g)Hn(f) for group homomorphisms f : G ---7 G' 9 G' ---7 Gil,

• Hn(f) is the identity homomorphism if f is the identity.

The following commands compute H 3 (f) : H 3 (P, Z) ---7 H 3 (85 , Z) for the inclusion f : P <-t 8 5 into the symmetric group 8 5 of its Sylow 2-subgroup. They also show that the image of the induced homomorphism H 3 (f) is precisely the Sylow 2-subgroup of H 3 (85 , Z).

r--------------------------- GAP ---------------------------,

I gap> S_5:~SymmetricGroup(5);; P:~SylowSubgroup(S_5,2);;

iThere is an analog of the restriction for homology which also satisfies this transitive property (Proposition 9.5 in Brown [BD.

190

gap> f:~GroupHomomorphismByFunction(P,S_S, x->x);; gap> R:~ResolutionFiniteGroup(p,4);;

gap> S:~ResolutionFiniteGroup(S_S,4);;

gap> zP_map:~EquivariantChainMap(R,S,f);;

gap> map:~TensorWithlntegers(ZP_map);;

gap> Hf:~Homology(map,3);;

gap> Abelianlnvariants(Image(Hf»; [2,4] gap> GroupHomology(S_S,3); [2,4,3]

If H is a subgroup of finite index in e then there is an analogous re­striction map in group homology (see for example Brown [B], §III.9).

7.2. Inflation

Let X* denote the bar resolution of e. Recall

Xn = EB(gl , ... ,gn)EGn R[g1, ... ,gn],

where the sum runs over all ordered n-tuples in en. If H is a subgroup of e, let xf denote the complex defined by

X:; = EB(glH, ... ,gnH)E(G/H)nR[g1H, ... ,gnH].

This is a resolution, and we have a chain map defined on n-cells by [g1, ... ,gn] f----+ [g1H, ... ,gnH].

Suppose that H is a normal subgroup of e and A is a e-module. We may view AH as a e I H-module. In this case, the homomorphism of pairs is given by the quotient map 0: : e ----7 e / H and the inclusion map f3 : A H ----7 A. The inflation map Inf is the induced map defined by (13), denoted

The inflation-restriction sequence in dimension n is

For a proof, see Weiss, [W], §3.4. There an analog of this inflation-restriction sequence for homology. We omit any discussion of transfer and Shapiro's lemma, due to space

limitations. Acknowledgements: I thank G. Ellis, M. Mazur and J. Feldvoss, P. Guil­

lot for correspondence which improved the content of these notes.

191

References

A. A. Adem, "Recent devolopments in the cohomology of finite groups," Notices AMS, vol 44(1997)806-812. Available online at http://www.ams.org/notices/199707/199707-toc.html.

Bi. M. Bishop, The GAP package CRIME, available from http://www.gap-system.org/Packages/crime.html

B. K. Brown, Cohomology of groups, Springer-Verlag, 1982. C. J. Carlson's page: http://www.math.uga.edu/-lvalero/cohointro.html. CEo E. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press,

1956. EM. S. Eilenberg, S. MacLane, "Relations between homology and homotopy

groups," Proc. Nat. Acad. Sci. U. S. A. 29 (1943). 155-158. E1. G. Ellis, The GAP package HAP 1.8.4, available from

http://www.gap-system.org/Packages/hap.html. E2. --, "Computing group resolutions", J. Symbolic Computation, 28 (2004),

1077-1118. available from http://hamilton.nuigalway.ie/preprints.html.

E. L. Evens, The cohomology of groups, Oxford Univ. Press, 1991. Gap. The GAP Group, GAP - Groups, Algorithms, and Programming, Ver­

sion 4.3, 2000 http://www.gap-system.org/.

G. R. Godement, Topologie algebrique et theorie des faisceaux, Hermann, 1958.

Gr. D. Green's page: http://www.math.uni-wuppertal.de/-green/Coho_v2/.

Gu. P. Guillot's page:

http://www-irma.u-strasbg.fr/-guillot/research/cohomology_of_groups/index.html

Ho. G. Hochschild, "Relative homological algebra," Trans. Amer. Math. Soc. 82 (1956), 246-269.

K. G. Karpilovsky, The Schur multiplier, Oxford Univ. Press, 1987. Ro. M. Roder, The GAP package HAPcryst, available from

http://www.gap-system.org/Packages/undep.html. R. J. Rotman, Advanced modern algebra, Prentice Hall, 2002. S. J.-P. Serre, Local fields, Springer-Verlag, 1979. Sh. S. Shatz, Profinite groups, arithmetic, and geometry, Princeton Univ.

Press, 1972. St. William Stein, SAGE Mathematics Software (Version 2.9), The

SAGE Group, 2007, http://www.sagemath.org.

W. E. Weiss, Cohomology of groups, Academic Press, 1969.

Doubles of Residually Solvable Groups

Delaram Kahrobaei

Mathematics Department, New York City College of Technology The City University of New York, 300 Jay Street, Brooklyn, NY 11201

CUNY Graduate Center, Doctoral Program in Computer Science 365 Fifth Avenue, New York, NY 10016

E-mail: dkahrobaei@citytech.cuny.edu

Abstract: In this paper we study residual solvability of doubles of residually solvable groups. We find suitable conditions where this kind of structure is residually solvable, and show that in general this is not the case. However this kind of structure is always meta-residually-solvable. *

Keywords: residually solvable, solvable separability, doubles

1. Introduction and Motivation

The notion of residual properties was first introduced by Philip Hall in 1954 (see [9], page 349). Let X be a class of groups. Gis residually-X if for every non-trivial element 9 in G there is an epimorph of G in X such that the element corresponding to 9 is not the identity. In this paper we consider the question of residual solvability of amalgamated products of residually solvable groups. That is when the free product with amalgamation of two residually solvable groups again residually solvable.

The question of being residually solvable can be simplified to whether the group is simple. Neumann in [16] asks the following question: Is it possible that the free product with amalgamation {A * Bj H = K}, where A, B are free groups of finite ranks, H, K are finitely generated subgroups of A,B, respectively, is a simple group? For the case where the amalgamated subgroup is not finitely generated, Ruth Camm [7] constructed an example of a simple free product with amalgamation G = {A * Bj H = K} where A, B are free groups of finite rank and their subgroups H, K have infinite rank. This example can be thought as amalgamated product of two residually

*2000 Mathematics Subject Classification. Primary 20E06.

192

193

solvable groups that is not residually solvable. (Le. where the amalgamated subgroup fails the maximal condition). In fact she showed that there exist continuously many non-isomorphic simple amalgamated products of two finitely generated free groups with non-finitely generated amalgamation.

For the case where the amalgamated subgroup is finitely generated Burger and Mozes [6] constructed an infinite family of torsion free finitely presented simple groups isomorphic to an amalgamated product {F * F; G} where F, G are finitely generated free groups.

Note that a partial answer to the question in [16], is that under the same conditions of the problem, {A * B; H = K} is not simple provided either of indices [A: H], [B : K] is infinite (see [10]).

By a double of a group A we mean an amalgamated product G =

{A * A; e = C} where - is an isomorphism from A onto itself. Clearly, doubles of free groups are never simple. In this paper, we study

residual solvability of amalgamated products of finitely generated residu­ally solvable groups. Since free groups are residually solvable, what we prove here is stronger than the corresponding result for the class of free groups. We note that any free product of residually solvable groups is residually solvable again [8]. Here we focus attention on the class of residually solv­able groups and show that the doubles of residually solvable groups are residually solvable if we impose the condition of solvable separability over the amalgamating subgroup:

Theorem 1.1. Let" - " be an isomorphism from a group A onto itself, e be a subgroup of A, and G be the amalgamated product of A and A amalgamating e with C, that is

G = {A * A; e = C}. If A is residually solvable and e is solvably separable in A then G is also

residually solvable.

In general we show that doubles of residually solvable groups are not residually solvable; note that these are one of the only non-trivial (simple or perfect groups) known non-residually solvable groups. However they are meta-residually solvable (see Corollary 3.1). Here is the statement of the

Theorem:

Theorem 1.2. Let A be a finitely generated residually solvable group, and e a normal subgroup of A, such that A/e is perfect. Let ,,- " be an iso­

morphic mapping of A onto itself. Then

D = {A * A; e = C}

194

is meta-residually solvable, but not residually solvable.

Here is an example of Theorem 1.2. Let A be a free group of rank two, there is a natural homomorphism from A onto A5 which is simple. Let e be the kernel of such a homomorphism. Clearly Ale is simple (perfect). Forming D = {A * .4; e = C} gives the example.

Trivially, solvable groups are residually solvable, and it is known that free groups are residually solvable. The author has studied residual solv­ability of generalized free products of finitely generated nilpotent groups (see [11], [12]), and proved that such structures are often residually solv­able. Such amalgamated products occur in one-relator groups (see [4]) and studying residual solvability of generalized free products could be also fruit­ful in answering old questions on residual solvability of one-relator groups. Studying residual solvability of groups was pioneered by work of Baumslag in 1971, when he proved positive one-relator groups are residual solvable (see [3]). Residual solvability of non-positive groups has been studied in very few cases, for example Kropholler in [13J shows that Baumslag-Solitar groups are free-by-solvable showing that the second derived subgroup is free, therefore it is residually solvable. In [1], Arzhantseva, de la Harpe to­gether with the author define the full prosolvable completion of a group, and these theorems have potential applications to the natural questions that arise.

Acknowledgment

I thank my Ph.D. supervisor G.Baumslag and also K.J.Falconer for helpful comments. The research of this author has been supported by CUNY re­search foundation at the City College of the City University of New York and New York Group Theory Cooperative.

2. Preliminaries

In this section, we recall some facts and prove some lemmas to be used later on.

2.1. Subgroups of amalgamated products

We will use a theorem by Hanna Neumann [15] extensively in this paper. With regard to abstract groups, Hanna Neumann showed in the 1950s that, in general, subgroups of amalgamated products are no longer amalgamated products, but generalized free products, indeed she proved the following:

195

let K be a subgroup of G = {A * B; C}, then K is an HNN-extension of a tree product in which the vertex groups are conjugates of subgroups of either A or B and the edge groups are conjugates of subgroups of C. The associated subgroups involved in the HNN-extension are also conjugates

of subgroups of C. As a corollary, if K misses the factors A and B (i.e.

K n A = {I} = K n B), then K is free; and if K misses the amalgamated

subgroup C (i.e. K n C = {l}), then K = I1iE/Xi * F, where the Xi are conjugates of subgroups of A and Band F is free (see [5] for more information) .

Later a description was given by the Bass-Serre theory [17], with groups acting on graphs to give some geometric intuition: the fundamental group of a graph of groups generalizes both amalgamated products, HNN-extensions and tree products.

2.2. Some Lemmas

Here we prove some lemmas to be used in proving the main results of this paper.

Lemma 2.1. If A is a g'rOUP, C is a subg'rOup of A, c/J is an isomorphic mapping of A onto a g'rOUp B, and D is the amalgamated p'rOduct of A and B amalgamating C with Cc/J, that is

then there is a homomorphism, 1jJ, f'rOm D onto one of the factors, and the

kernel of 1jJ, K, is:

K = gp(a(ac/J)-lJa E A).

Furthermore this map is injective on each factor.

Proof. Let a be the homomorphism from A onto itself, and f3 be the homomorphism from B onto the inverse of the isomorphic copy of A, i.e. f3 = c/J-l. These homomorphisms can be extended to a homomorphism from D onto A, ([5] page 103, [14]). Since 1jJ(Cab) = Ca(bc/J-l) it follows easily

K = ker1jJ = gp(a(ac/J- 1 )Ja E A). By the way that a and f3 are defined, it follows that this homomorphism is one-to-one restricted to either A or B.

o

Lemma 2.2. Let A, B, C, D, K and c/J be as in Lemma 2.1 with C a p'rOper subg'rOup of A. Then K is not central in D, in other words, [K,D] i- {1}.

196

Proof. First, note that the center of Dis eD = eA n eB n C. By Lemma 2.1, KnA = {I}, and hence eAnK = {I}. This implies that [K,DJ # {I}.

o

Lemma 2.3. Let A, B, C, D, K and ¢ be as in Lemma 2.1. Furthermore, let C be normal in A. Then K commutes with C, i.e. [C, KJ = 1.

Proof. Let c E C, and a E A. Since C <l A and C¢ <l B, and ¢ is an isomorphism, we can do the following computation:

(a-1ca)¢ = (a-1¢)(c¢)(a¢) = (a-1¢)c(a¢).

Note that a(a¢)-l E K, so

ca(a¢)-l = a(a¢)-lc;

that is [C,K] = {I}. o

Corollary 2.1. Let A, B, C, D, K and ¢ be as above. Then K is free.

Proof. Since the homomorphism 'ljJ is one-to-one restricted to either A or B, then

KnA = {I} = KnA¢.

So, by the theorem of Hanna Neumann mentioned in Subsection 2.1, K is free. 0

2.3. The filtration approach to residual solvability

In this section we provide some background for filtration approach which we will use later.

A family (A>.I'\ E A) of normal subgroups of A is termed a solvable filtration of A if AI A>. is solvable for every ,\ E A and n>'EA A>. = {I}. We shall say that H is solvably separable in A if n~=l H A>. = H, that is if and only if n~=l A>. c H. Now let H ~ A, then (A>.I'\ E A) is called an H-filtration of H if n>'EA H A>. = H.

Let ¢ : H ---t K be an isomorphism between subgroups H of A and K of B. Two equally indexed filtrations (A>. 1,\ E A) and (B>.I'\ E A) of A and B respectively are termed (H, K, ¢)-compatible if (A>.nH)¢ = B>.nK (\f'\ E

A). The following Proposition of Baumslag [2] will help us to prove one of the results: let (A>.J'\ E A), (B>.JA E A) be solvable (H, K, ¢)-compatible filtrations of the residually solvable groups A and B respectively. Suppose

197

(A>JA E A) is an H -filtration of A and (B>.IA E A) is a K-filtration of B. If, for every A E A,

{A/A>. * B/B>.;HA>./A>. = KB>./B>.},

is residually solvable, then so is G = {A * B; H = K}.

3. Doubles of residually solvable groups

In this section we prove the theorems concerning the doubles of residually solvable groups.

3.1. M eta-residual-solvability

Let X be a group property. Then a group G is meta-X if there exist A and Q of property X and a short exact sequence 1 ----+ A ----+ G ----+ Q ----+ 1.

Here we prove that in general the amalgamated products of doubles of residually solvable groups are meta-residually-solvable.

Proposition 3.1. Let A be a residually solvable group, C be a subgroup of A, and 11- II be an isomorphic mapping of A onto A. Then the generalized free product of A and A amalgamating C with C,

G = {A * A; C = C}

is an extension of a free group by a residually solvable group.

Proof. Let <jJ : G ----+ A; then K = ker<jJ = gp(aa-1Ia E A). K is free by the theorem of Hanna Neumann mentioned in subsection 2.1, since

AnK = {1} = Anker<jJ.

Therefore G is an extension of a free group by a residually solvable group.

o

Corollary 3.1. G is meta-residually-solvable.

Proof. Since free groups are residually solvable, then by Proposition 3.1, G is residually solvable-by-residually solvable. That is to say that G IS

meta-residually-solvable. o

3.2. Effect of solvably separability on the amalgamated subgroup and residual solvability, Proof of Theorem 1.1

We now prove that if we impose the solvable separability condition on the amalgamated subgroup of doubles of residual solvable groups then the resulting group is residually solvable.

198

Proof. Assuming C is solvably separable in C and A is residually solvable, we want to show that G is residually solvable. That is we must show that for every non-trivial element (1 i=)d E G, there exists a homomorphism, ¢, from G onto a solvable group S, ¢ : G ----) S, such that d¢ i= 1. We consider two cases: Case 1: Let 1 i= d E A. There exists an epimorphism ¢ from G onto A, so that d¢ = d. Since A is residually solvable, there exists .A E N, such that d rf. oAA, where oAA, is the .A-th derived group of A. Now put S = Ajt5AA, a solvable group of derived length at most .A. Note that the canonical homomorphism, (), from A onto S, maps d onto a non-trivial element in S. Now consider the composition of these two epimorphisms, ()o¢, which maps G onto S. The image of d in S is non-trivial:

() 0 ¢(d) = d() = doAAi=s1.

Case 2: Let 1 i= d rf. A but d E G. Now d can be expressed as follows:

d=albla2b2···anbn (aiEA-C biEA-C).

Since the equally indexed filtrations, {oAAhEN' and {oAAhEN of A and A are compatible, we can form GA:

Note that GA is residually solvable (by mapping it to one of the factors and noting that the kernel of the map is free). Consider the canonical homomorphism () from G onto GA. Since C is solvably separable in A, i.e.

n CoAA = C, AEN

.A E N can be so chosen that

ai rf. CoAA, ai rf. CoAA (for i = 1,··· ,n).

Hence

aloAA b1oAA··· anoAAbn oAAi=c>.1.

This completes the proof of theorem by using Baumslag's Proposition [2], we recalled in Section 2.3. 0

Note also, if G is residually solvable then C is solvably separable in A, since solvable separability is equivalent to n~l H AA C C.

199

3.3. Solvable separability is a sufficient condition for residual solvability; Proof of Theorem 1.2

Note that the condition of solvable separability of the amalgamated sub­group in the factors, in the case of doubles, is necessary. The following theorem shows that the amalgamated product of doubles is not residually solvable where the factors are residually solvable groups.

Proof. D is meta-residually solvable by Corollary 3.1.

By Lemma 2.1, there exists an epimorphism from D onto A. Let K be the kernel of this epimorphism. Since G is normal in A, by using Lemma 2.3

Now we want to show that D is not residually solvable. We proceed by con­tradiction. Suppose D is residually solvable. Let d be a non-trivial element in [K, D]. The existence of such an element is guaranteed by Lemma 2.2. Now assume fJ, is a homomorphism of D onto a solvable group S, so that dfJ, =/=8 1. Since fJ, is an epimorphism, GfJ, is a normal subgroup of S, and by (* ),

If we can show that S = GfJ" then [KfJ" S] = 1, which implies that dfJ, =81, a contradiction. We now need to show that S = G fJ,. We have that D -7> S induces a homomorphism from D/G onto S/GfJ,. Since A/G and A/e each have a perfect subgroup, this induces a homomorphism from A/G to 1. So,

S/GfJ, = 1 and hence GfJ, = S. 0

References

1. G.Arzhantseva,P. de la Harpe and D.Kahrobaei The true prosoluble comple­tion of a group: Examples and open problems, Journal of Geometiae Dedicata, Springer Netherlands 124 (2007), 5-26.

2. G.Baumslag, On the Residual Finiteness of Generalized Free Products of Nilpotent Groups, Trans. Amer. Math. Soc 106 (1963), 193-209.

3. G.Baumslag, Positive One-Relator Groups, Trans. Amer. Math. Soc 156 (1971), 165-183.

4. G.Baumslag, A Survey of Groups with a Single Defining Relation, London Math. Soc. Lecture Note Ser. Camb. Univ. Press, Proc. of Groups St Andrews 121 (1986), 30-58.

5. G.Baumslag, Topics in Combinatorial Group Theory, Birkhauser-Verlag, 1993.

6. M.Burger and S, Mazes, Finitely Presented Simple Groups aqnd Products of Trees, C.R. Acad.Sci. Paris Ser. 1 Math. 7 (1997), 747-752.

200

7. R.Camm, Simple Free Products, J. London Math. Soc. 28 (1953), 66-76. 8. K.W. Gruenberg, Residual Properties of Infinite Solvable Groups, Proc. Lon­

don Math. Soc. 7 (1954), 29-62. 9. P.Hall, The Splitting Properties of Relatively Free Groups, Proc. London

Math. Soc. 4. (1978), 343-356. 10. D.Kahrobaei, A Simple Proof of a Theorem of Karrass and Solitar, Contemp.

Math. Soc. 372. (2005), 107-108. 11. D.Kahrobaei, On Residual Solvability of Generalized Free Products of Finitely

Genertaed Nilpotent Groups,J. Group Theory (accepted) (2007). 12. D.Kahrobaei, Residual Solvability of Generalized Free Products, PhD Thesis,

CUNY Graduate Center (1994) 13. P.H. Kropholler, Baumslag-Solitar Groups and Some Other Groups of Coho­

mological Dimension Two, Comment. Math. Relv. 65 (1990), 547-558. 14. B.H. Neumann, An Essay on Free Products of Groups with Amalgamation,

Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503-554. 15. H. Neumann, Generalized Free Products with Amalgamated Subgroups. ii,

Amer. J. Math. 71 (1949), 491-540. 16. P. Neumann, SQ-universality of Some Finitely Presented Groups, J. Aust.

Math. Soc. (1973). 17. J.P. Serre, Trees, Springer-Verlag, (1980).

AN APPLICATION OF A GROUP OF OL'SHANSKII TO A QUESTION OF FINE ET. AL.

Seymour Lipschutz

Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122

Dennis Spellman

Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122

Dedicated to Tony Gaglione on his Sixtieth birthday.

1. Introduction

The classical commensurator of a subgroup may be found in [KM] where it is called the virtual normalizer. Proposition 2.4 of [BBFGRS] asserts that any nontrivial finitely generated subgroup H -=I=- {I} in a free group F has finite index in its commensurator. Following the above proposition the authors of [BBFGRS] define a group G satisfying the commensurator condition as one for which every finitely subgroup H -=I=- {I} has finite index in its commen­surator. The authors of [BBFGRS] asked for a finitely generated infinite group G with the commensurator property that is not hyperbolic. In [0] Ol'shanskii constructed a 2-generator infinite simple group with the prop­erty that every nontrivial proper subgroup is infinite cyclic. We show that any such group satisfies the commensurator condition but is not hyperbolic. In [BBFGRS] the authors define for finitely generated groups a condition called property R. We give a definition, which coincides with theirs in the case of finite generation, but which applies to arbitrary groups. We consider operators which preserve property R as well as those that do not; moreover, we reflect upon the relationships (or lack thereof) between property Rand various other finiteness conditions. Finally we introduce a generalization of property R which we call weak property R or property WR. The paper is divided into six sections beginning with this Introduction (Section 1). In Section 2 we introduce the commensurator condition. In Section 3 we recall

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various conditions on maximal abelian subgroups and apply these to argue that Ol'shanskii's group is an example of the type we seek. In Section 4 we study property R. In Section 5 we introduce property WR. In Section 6 we pose some questions.

2. The Commensurator Condition

In this section we introduce the commensurator of a subgroup and the commensurator condition.

Definition 2.1. Let G be a group and H a subgroup in G. The commen­surator of H in G, denoted by commc(H), consists of the elements 9 E G

such that

IH: H n H91 < 00 and IH9 : H n H 91 < oo}.

One can easily prove:

Proposition 2.1 (BBFGRS). : If H is a subgroup in the group G, then commc(H) is a subgroup in G.

Clearly H is a subgroup of commc(H).

Proposition 2.2 (BBFGRS). If F is a free group then, for every non­trivial finitely generated subgroup H =I- {ll in F, one has [commF(H) : H] < 00.

Now we define the commensurator condition.

Definition 2.2 (BBFGRS). A group G satisfies the commensurator condition provided, for every nontrivial finitely generated subgroup H =I­{l} G one has [commc(H) : H] < 00.

Thus free groups satisfy the commensurator condition. The reason this rather technical condition is of some interest is that it implies a finiteness condition, Stalling's Property S, that has been studied in [BBFGRS].

3. Conditions on Maximal Abelian Subgroups and the Counterexample

If 9 is an element of the group G let Gc(g) = {x E G : gx = xg} be its centralizer in G. In this section we state some definitions, which we deem sufficiently classical to omit references to the literature.

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Proposition 3.1. Let G be a group. The following three conditions are pairwise equivalent.

(1) The relation commutes with is transitive on G \ {1}. (2) Gc(g) is abelian for every 9 =I- 1 in G (3) For every pair M1 =I- M2 of distinct maximal abelian subgroups in

G one has M1 n M2 = {1}.

Definition 3.1. A group satisfying anyone (and therefore all three) ofthe conditions of Proposition 3.1 is commutative transitive or CT.

Definition 3.2. A subgroup M in a group G is malnormal in G provided 9 E G \ M implies g-l M 9 n M = {1}.

Definition 3.3. A group G such that all its maximal abelian subgroups are malnormal is said to be conjugately separated abelian or CSA .

Remark 3.1. Every abelian group is trivially both CT and CSA.

Proposition 3.2. Every GSA group is GT.

Proof. Suppose G is CSA. Let M1 and M2 be distinct maximal abelian subgroups and assume to deduce a contradiction that 9 =I- 1 lies in M 1 nM2 .

Let m E M 1 . Then g = m-1gm lies in m- 1 M 2m n M 2. Since Gis CSA this implies m E M 2 . Therefore, since m E M1 was arbitrary, M1 is a subgroup of M2. By maximality M1 = M2, contrary to hypothesis. Thus, M1 =I- M2 implies M1 n M2 = {1} whenever M1 and M2 are maximal abelian in G. Hence, G is CT whenever it is CSA. 0

Remark 3.2. The infinite dihedral group D =< a, b; a2

b- 1 > is CT but not CSA.

To see this note that a free product of abelian groups is CT. On the other a CSA group cannot contain a nontrivial abelian subgroup. However in D the commutator subgroup is cyclic and hence D cannot be CSA.

Proposition 3.3. (Myasnikov and Remeslennikov): Let G be a group such that Gc(g) is infinite cyclic for all 9 =I- 1 in G. Then G is GSA.

Proof. If G = {1} then the result is trivially true. Now assume that G =I­{1} satisfies the hypotheses of the proposition and observe that G must be torsion free and CT. Let M =< m > be maximal abelian in G where, m =I- 1 and is not a proper power in G. Suppose to deduce a contradiction

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that 9 E G \ M but g-1Mg n M i= {I}. Then 9 does not commute with m. In particular, 9 i= 1. Since G is CT and the maximal abelian subgroups M and g-1 Mg intersect nontrivially we must have g-1 M 9 = M and the inner automorphism x ---> g-1xg of G restricts to an automorphism of M. Hence, g-1mg = m or g-1mg = m- 1 . But we are assuming that 9 does not commute with m so, g-1mg = m-1. Then g-2mg2 = g-1m- 1g = m and

g2 commutes with m. g2 i= 1 since G is torsion free. But 9 commutes with g2 ,which commutes with m, so 9 commutes with m since G is CT. This contradicts 9 1:. M = Cc(m). The contradiction shows that the condition g-1 Mg n M i= {I} implies gEM. Thus G is CSA. 0

In [BBFGRS] it was asked to give an example of finitely generated group that is neither free nor hyperbolic but which satisfies the commensurator condition. We show that a group constructed by Ol'Shanskii provides such an example. In [OJ Ol'shanskii constructed a 2-generator infinite simple group with the property that every proper nontrivial subgroup is infinite cyclic. For the remainder of this section let G be one fixed such group, which shall be called Olshanskiis group. Since G is a nonabelian simple group it is centerless. Thus, for each 9 i= 1 in G the centralizer Cc(g) is proper in G (and nontrivial since 9 E Cc(g)) hence infinite cyclic. It follows from Proposition 3.1 that G is CSA. We claim that

Proposition 3.4. The group G satisfies the commensurator condition and further G is not hyperbolic.

Proof. We first show that G satisfies the commensurator condition. In order to do this we must demonstrate that, if H i= {I} is a finitely generated subgroup in G, then [commc(H) : HJ < 00. First of all ob­serve that every subgroup of G is finitely generated so that the finite­ness condition is redundant. Note also that commc(G) = G and thus [commc(G) : G] = 1 < 00. So we must show, for every proper sub­group H i= {I} in G, [commc(H) : H] < 00. Now if commc(H) were infinite cyclic wed be finished since every nontrivial subgroup of an infi­nite cyclic group has finite index in the containing group. In other words it suffices to show that commc(H) is a proper subgroup of G when­ever H i= {I} is proper in G. Now if H i= {I} is proper in G it is cyclic so H extends to a maximal abelian subgroup M. Let 9 E G \ M. We claim that 9 1:. commc(H). Suppose to deduce a contradiction that 9 E commc(H). Then, in particular, [H : g-1 H g7 H] < 00. This implies that g-1Hg n H i= {I} since H is infinite cyclic. But then g-1Mg n M

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g-l H 9 n H -=I=- {1} and so gEM since G is eSA. That contradicts the choice of g. The contradiction shows 9 tf. commc(H). Hence, commc(H) is proper in G whenever H -=I=- {1} is proper in G and we are finished. This shows that G satisfies the commensurator condition. 0

To show that G is not hyperbolic we need the following definition.

Definition 3.4. (Fine and Rosenberger [FR]) A group H is restricted Gromov or RG provided, if 9 and h are any elements in H, then either the subgroup < g, h > is cyclic or there exists a positive integer t with gt -=I=- 1, ht -=I=- 1 and < gt, ht > is the free product < gt > * < ht >.

We state without proof

Proposition 3.5. (Gromov): Torsion free hyperbolic groups are RG.

Proof. We can deduce that G is not hyperbolic by showing that Olshan­skiis group G is not RG. Suppose that a and b generate the 2-generator group G. If G were RG there would exist a positive integer t such that < at, bt >=< at > * < bt >. Now < at > and < bt > are each infinite cyclic since G is torsion free. Thus, < at, bt > is free of rank 2. Since every proper subgroup of G is cyclic we conclude G =< at, bt > is free of rank 2 an obvious contradiction. The contradiction shows that G is not hyperbolic completing the proposition. 0

4. Property R

Definition 4.1. A group G satisfies Property R provided, for every finite subset X ~ G and every subgroup H ~< X >, it is the case that H has finite index in < X > if and only if, for each x EX, there is a positive integer n(x) such that xn(x) E H.

Definition 4.2. A group G satisfies Stallings Property S if for any finitely generated subgroups A and B of G with [A : AnB] < 00 and [B : AnB] < 00 we have [< A, B >: An B] < 00, that is for any two commensurable finitely generated subgroups in G the intersection must have finite index in the join.

Definition 4.1 coincides with that given in [BBFGRS] in the case G is finitely generated the only case for which property R was defined in [BBF­GRS]. Property R is of interest because it implies property S and because

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there are many examples. Every nilpotent group satisfies property R (Mal­cev [M]) as does every direct product G x F where G satisfies property R and F is finite [BBFGRSj.

The question was asked whether there exists a group G satsifying Prop­erty R but not virtually nilpotent. Recall that if P is a group property then G is virtually-P if G has a subgroup of finite index which satisfies P. In [BBFGRS] the following example was given which answered the question.

Example 4.1 (BBFGRS). : Let n ~ 2 be an integer. Then the group En =< a, t; rlat = an > satisfies properly R but is not virlually nilpotent.

The following proposition provides another example of such a group.

Proposition 4.1. Ol'Shanskii's group G satisfies Properly R and is not virlually nilpotent,

Proof. Let X be a finite subset of Olshanskiis group G. If < X > is cyclic then it is nilpotent so satisfies property R. Thus to show that G satisfies property R it suffices to take a finite subset X = Xl" Xk which generates G. We may assume without loss of generality that 1 tt X. Now suppose that H is a subgroup of infinite index in G. Thus, H is a proper subgroup of G. Now if x;(j) E H where n(j) is a positive integer, j = 1" k, then

the x~(j) commute in pairs since H, being proper, is cyclic. Moreover, since

G is torsion free and no Xj = 1, we must have x;(j) -:j 1, j = 1" k. By commutative transitivity the Xj must commute in pairs. But, since the Xj generate G, that would imply that G is abelian a contradiction. The contradiction shows that G satisfies property R. It is an easy exercise to show that the only subgroup of finite index in an infinite simple group is the entire group. No nonabelian simple group can be nilpotent. Thus, Ol'shanskii's group is another example of a property R group which is not virtually nilpotent. 0

Example 4.2. Let A =< a, b; ab = ba > be free abelian of rank 2 with basis {a, b}. A certainly satisfies property R. Let T E Aut( A) be determined by transposing a and b. Let

G =< a,b,t;ab = ba,Clat = b,Clbt = a,t2 = 1 >

be the semidirect product determined from this data. Let X = {a, t}. Since b = rlat, X generates G. Let H =< a >. Then H has infinite index in G =< X >. Indeed, H =< a > has infinite index in the subgroup

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A =< a, b >. But al = a E Hand t 2 = 1 E H. Thus G violates property R.

The authors of [BBFGRS] asked for a characterization of those finite extensions of property R groups which retain property R. We believe that, in some sense, the direct product result is best possible since the semidirect product of Example 4.2 violates property R.

Proposition 4.2. Property R is preserved in subgroups, homomorphic im­ages and direct limits.

Proof. It is obvious from our definition of property R that it is inherited by subgroups. Suppose next that G satisfies property Rand ¢ : G ----> G' is an epimorphism. Let Y = {YI" yd be a finite subset of G' and suppose H is a subgroup of < Y >. Suppose further that there are positive integers n(j) such that Y7(j) E H, j = 1" k. We want to show that H has finite index in < Y >.

For each 1 :<:; j :<:; k fix a preimage Xj E G of Yj under ¢. We claim that

U(YI"Yk)V(YI"Yk)-1 ~ H

implies that

If not then

U(YI"Yk)V(YI"Yk)-1 E< y~(l)"y~(k) >c H.

Thus, if there were infinitely many distinct right cosets of H in < Y >, there would be infinitely many distinct right cosets of < X~(l)" x~(k) > in < Xl, ,Xk >. This contradicts the fact that G satisfies property R. The contradiction shows that [< Y >: H] < 00. Hence, G' satisfies property R and property R is preserved under homomorphic images.

Finally suppose that A is an upward directed set and (G.x).xEA is a family of property R groups indexed by A. Suppose that, for each ,J1, E A with >. < J1, there is a homomorphism P.x,!, : G.x ----> G and suppose further that, for each >., J1" 1/ E A with>' < J1, < 1/, one has p.x,v = P.x,!'P!',v where we write our maps to the right of their arguments and compose accordingly. Let G be the direct limit determined from this data and, for each>' E A, let P.x : G.x ----> G be the limit map. It will suffice to show that, if r = hI, ... , ,d is any finite subset of G, then the subgroup of G generated by r satisfies property R. Fix such a subset r. Then there is >. E A and gl, ... ,gk E G.x

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such that "/j = gjP>', j = 1" k. Now < "/1, ... , "/k > is a subgroup of the homomorphic image G>.p>.. Hence, < "/1, ... , "/k > satisfies property R. 0

Let G be a finitely generated group. We wish to examine the relationship between property R and various other finiteness conditions for G. Specifi­

cally we consider (1) The maximum condition. (2) The minimum condition. (3) Residual finiteness. ( 4) Hopficity.

The following will be useful for our purposes.

Lemma 4.1. Let G be an infinite group generated by finitely many torsion

elements. Then G does not satisfy property R.

Proof. Suppose X = {Xl, ... , xd generates G and x?(j) = 1 where each

n(j) 2: 2,j = 1, ... , k. Then H = {I} has infinite index in G yet x?(j) E

H,j = 1, ... ,k. 0

For example, the infinite dihedral group presented as D =< a, b; a2 =

1, a-lba = b- l > is generated by {a, ab} and each of a and ab has order 2 in D. Thus D violates property R. In the same paper in which Ol'shanskii constructed the group in Section 3 he showed how to modify the construc­tion to create a 2-generator infinite group in which every proper subgroup is finite cyclic. Since such a group clearly satisfies both the maximum and minimum conditions neither one (nor both) implies property R as a conse­quence of Lemma 4.1 applied to Ol'shanskii's modified construction. Con­versely property R implies neither condition. The groups B N of Exam­ple 4.1 violate the maximum condition as the normal closure of < a > is isomorphic to the additive group of the ring Z[~l and is not finitely generated as a group. The infinite cyclic group, for example, satisfies prop­erty R but violates the minimum condition. The infinite dihedral group D =< a, b; a2 = 1, a-1ba = b-1> is easily seen to residually be in the fam­ily of finite dihedral groups Dn =< a, b; a2 = 1, bn = 1, a-1bna = bn- 1 > of order 2n. Hence, D is a finitely generated residually finite group and thus also Hopfian. This shows that the Hopf property or even residual finiteness does not imply property R. On the other hand Ol'shanskii's group of Sec­tion 3 is not residually finite as the only finite image is the trivial group {I}. Ol'shanskii's group is, however, Hopfian as it is easy to see every simple group must be.

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Proposition 4.3. Property R is not preserved in unrestricted direct prod­ucts or even unrestricted direct powers.

Proof. Let N be the set of positive integers. Since the infinite dihedral group D is residually in the family {Dn : n E N} of finite dihedral groups, D embeds in the unrestricted direct product IInENDn. If IInENDn satisfied property R then so would the subgroup D a contradiction. Each Dn, being finite, satisfies property R. Let Soo, the infinite symmetric group, be the group of all permutations of N which move only finitely many integers. For each n E N fix an embedding of Dn into Soo. Now Soo satisfies property R since it is the direct limit of the family {Sn : n E N} of finite symmetric groups. But IInENDn embeds in the unrestricted direct power S!. Hence, S! violates property R. D

5. Weak Property R

Proposition 5.1. The following conditions on a group G are equivalent. (WR1) If Go is any subgroup of G and Go* is any homomorphic

image of Go, then the set of torsion elements in Go* forms a locally finite

subgroup of Go*. (WR2) If X is any finite subset of G and N is any normal subgroup

of < X >, then N has finite index in < X > if and only if for each x E X there is a positive integer n(x) such that xn(x) EN.

Note that the set of torsion elements in a property S group forms a locally finite subgroup. However, property S is not in general preserved in homomorphic images. This latter fact can be seen from the result that free groups satisfy property S.

Proof. (of Proposition 5.1)

WR1 ===} WR2

Assume G satisfies WRl. Let X = {Xl, ... , xd be a finite subset of G and let Go =< X >. Let N be a normal subgroup of Go. Suppose that for j = 1, ... , k there is a positive integer n(j) such that x;(j) E N. Then {N Xl, ... , N Xk is contained in the set T of torsion elements of the homo-morphic image GaiN of Go. Thus, GaiN =< NXI, ... , NXk > is a finite group and so [< X >: N] = [Go: N] < 00. Hence G satisfies WR2.

WR2 ===} WR1

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Suppose G satisfies WR2. Let Go be a subgroup of G and let ¢ : Go -? Go* be an epimorphism. It will suffice to show that, if T is the set of torsion elements in Go* and (91" 9k) E Tk is a finite tuple of elements ofT, then the subgroup < 91" 9k > of Go* is finite. For each j = 1" k choose a preimage Xj E Go of 9j under ¢. Let H =< X1"Xk > nKer(¢). Now since each Xj

maps into T there is a positive integer n(j) such that x7(j) E H, j = 1, ,k. Furthermore, letting X = X1"Xk, H is normal in < X > since Ker(¢) is normal in Go and H =< X > nKer(¢). Since G satisfies WR2 we must have that H has finite index in < X >. Then

< X > j H = < X > j ( < X > nK er( ¢))

= « X> Ker(¢)jKer(¢) =< X1¢"Xk¢ >=< gl"gk >

is finite. Hence G satisfies WRl. o

Definition 5.1. A group satisfies weak property R or property WR pro­vided it satisfies either one (and therefore both) of the conditions WR1 or WR2 of Proposition 5.1.

Definition 5.2. A group is a U-group if roots, when they exist, are unique.

Definition 5.3. A group is an FC-group provided every conjugacy class is finite.

Proposition 5.2. (B.H. Neumann [N}): A torsion free FC-group is abelian.

Proposition 5.3. Let G be a torsion free property WR group. Then Gis aU-group.

We note that both property WR1 and the fact that torsion freeness implies uniqueness of roots within the category are well-known properties of nilpotent groups.

Proof. (of Proposition 5.3) Suppose that G is a torsion free property WR group. Suppose n is a positive integer and x, y E G satisfy xn = yn. We must show that x = y. We may assume without loss of generality that G =< x, Y > since property WR is clearly inherited by subgroups.

Let xn = Z = yn. Then z is central in G so < z > is normal in G. Moreover, since xn = yn = z and {x,y} generates G we must have [G :< z >J < 00 as G satisfies property WR. Now let w E G be arbitrary. Then

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< z ><:;;; Gc(w) and so from [G :< z >] = [G : Gc(w)][Gc(w) :< z >] we see that [G: Gc(w)] < 00. It follows that G is an Fe-group. By Proposition 5.2, G is abelian. But then, in the torsion free abelian group G, xn = yn implies (xy-l)n = 1 which in turn implies x = y. 0

We note that a slight variation of the proof shows that torsion free property S groups are U-groups.

6. Questions

Question 6.1. Must every finitely generated property R group be Hopfian?

Question 6.2. Are property R groups closed under finite direct products? Are they closed under finite direct powers?

Question 6.3. Does property WR imply property R?

Question 6.4. Must every torsion free property WR group embed in a property WR group admitting roots? What about torsion free property R groups? What about torsion free property S groups?

7. References

[BBFGRS] G. Baumslag, O. Bogopulski, B. Fine, A.M. Gaglione, G. Rosen­berger and D. Spellman, On some finiteness properties in infinite groups Alg. Colloquium, 15(1), 2007, 1-22

[FR] B. Fine and G. Rosenberger, On restricted Gromov groups Comm. In Alg., 20(8), 1992, 2171 2181

[KM] I. Kapovich and A.G. Myasnikov, Stallings foldings J. Alg. , 248, 2002, 608 668

[M] A.I. Malcev, Nilpotent torsion free groups Izv. Akad. Nauk. SSSR, 67, 1949, 347 366

[N] B.H. Neumann, Groups with finite classes of conjugate elements Proc. London Math. Soc., 3, 1951, 178 187

[0] A.Yu. Olshanskii, Infinite groups with cyclic subgroups Soviet Math. Dokl., 20(2), 1979,343 346

Quotient Isomorphism Invariants of a Finitely Generated Coxeter Group

Michael Mihalik

Mathematics Department, Vanderbilt University, Nashville TN 37240, USA

John Ratcliffe

Mathematics Department, Vanderbilt University, Nashville TN 37240, USA

Steven Tschantzk

Mathematics Department, Vanderbilt University, Nashville TN 37240, USA

1. Introduction

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Cox­eter groups. Coxeter [4]] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups.

In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W. Each of these invariants is the isomorphism type of a quotient group WIN of W by a characteristic subgroup N. The virtue of these invariants is that WIN is also a Coxeter group. For some of these invariants, the isomorphism problem of WIN is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups.

We emphasize that even if the isomorphism problem for finitely gener­ated Coxeter groups is eventually solved, several of the algorithms described in our paper will still be useful because they are computationally fast and would most likely be incorporated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups.

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In §2, we establish notation. In §3, we describe two elementary quoti­enting operations on a Coxeter system that yields another Coxeter system. In §4, we describe the binary isomorphism invariant of a finitely generated Coxeter group. In §5, we review some matching theorems. In §6, we describe the even isomorphism invariant of a finitely generated Coxeter group. In §7, we define basic characteristic subgroups of a finitely generated Coxeter group. In §8, we describe the spherical rank two isomorphism invariant of a finitely generated Coxeter group. In §9, we make some concluding remarks.

2. Preliminaries

A Coxeter matrix is a symmetric matrix M = (m(s, t))s,tES with m(s, t) either a positive integer or infinity and m(s, t) = 1 if and only if s = t. A Coxeter system with Coxeter matrix M = (m(s, t))s,tES is a pair (W,5) consisting of a group Wand a set of generators 5 for W such that W has the presentation

W = (5 I (st)m(s,t) : s, t E 5 and m(s, t) < 00).

We call the above presentation of W, the Coxeter presentation of (W, 5). If (W,5) is a Coxeter system with Coxeter matrix M = (m(s, t))s,tES, then the order of st is m(s, t) for each s, tin 5 by Prop. 4, p. 92 of Bourbaki [3], and so a Coxeter system (W,5) determines its Coxeter matrix; moreover, any Coxeter matrix M = (m(s, t))s,tES determines a Coxeter system (W, 5) where W is defined by the corresponding Coxeter presentation. If (W, 5) is a Coxeter system, then W is called a Coxeter group and 5 is called a set of Coxeter generators for W, and the cardinality of 5 is called the rank of (W, 5). A Coxeter system (W,5) has finite rank if and only if W is finitely generated by Theorem 2 (iii), p. 20 of Bourbaki [3].

Let (W,5) be a Coxeter system. A visible subgroup of (W, 5) is a sub­group of W of the form (A) for some A <;;; 5. If (A) is a visible subgroup of (W,5), then ((A),A) is also a Coxeter system by Theorem 2 (i), p. 20 of Bourbaki [3].

When studying a Coxeter system (W,5) with Coxeter matrix M it is helpful to have a visual representation of (W,5). There are two graphical ways of representing (W,5) and we shall use both depending on our needs.

The Coxeter diagram (C-diagram) of (W,5) is the labeled undirected graph ~ = ~(W, 5) with vertices 5 and edges

{(s,t): s,t E 5 and m(s,t) > 2}

such that an edge (s, t) is labeled by m(s, t). Coxeter diagrams are useful

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for visually representing finite Coxeter groups. If A c S, then ,6.( (A), A) is the sub diagram of ,6.(W, S) induced by A.

A Coxeter system (W, S) is said to be irreducible if its C-diagram ,6.

is connected. A visible subgroup (A) of (W, S) is said to be irreducible if «(A), A) is irreducible. A subset A of S is said to be irreducible if (A) is irreducible.

A subset A of S is said to be a component of S if A is a maximal irre­ducible subset of S or equivalently if ,6.( (A), A) is a connected component of ,6.(W, S). The connected components of the ,6.(W, S) represent the factors of a direct product decomposition of W.

The presentation diagram (P-diagram) of (W, S) is the labeled undi­rected graph r = r(W, S) with vertices S and edges

{(s,t): s,t E Sand m(s,t) < oo}

such that an edge (s, t) is labeled by m(s, t). Presentation diagrams are useful for visually representing infinite Coxeter groups. If A c S, then r( (A), A) is the sub diagram of r(W, S) induced by A. The connected com­ponents of r(W, S) represent the factors of a free product decomposition of W.

For example, consider the Coxeter group W generated by the four re­flections in the sides of a rectangle in E2. The C-diagram of (W, S) is the disjoint union of two edges labeled by 00 while the P-diagram of W is a square with edge labels 2.

Let (W, S) and (W', S') be Coxeter systems with P-diagrams rand r ' , respectively. An isomorphism ¢ : (W, S) ----> (W', S') of Coxeter systems is Hn isomorphism ¢ : W ----> W' such that ¢(S) = S'. An isomorphism 'Ij; : r ----> r ' of P-diagrams is a bijection from S to S' that preserves edges and their labels. Note that (W, S) ~ (W', S') if and only if r ~ r'.

We shall use Coxeter's notation on p. 297 of [5 for the irreducible spher­ical Coxeter simplex reflection groups except that we denote the dihedral group D~ by D2(k). Subscripts denote the rank of a Coxeter system in Coxeter's notation. Coxeter's notation partly agrees with but differs from Bourbaki's notation on p.193 of [3].

Coxeter [4]] proved that every finite irreducible Coxeter system is iso­morphic to exactly one of the Coxeter systems An, n ~ 1, B n, n ~ 4, Cn, n ~ 2, D 2 (k), k ~ 5, E 6 , E 7 , E s , F 4 , G 3 , G 4 . For notational convenience, we define B3 = A 3 , D2(3) = A 2, and D 2(4) = C 2

The type of a finite irreducible Coxeter system (W, S) is the isomorphism type of (W, S) represented by one of the systems An, B n, Cn, D 2 (k), E 6 ,

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E 7 , E s , F4, G 3 , G 4 · The type of an irreducible subset A of S is the type of ((A),A).

The C-diagram of An is a linear diagram with n vertices and all edge labels 3. The C-diagram of Bn is a V-shaped diagram with n vertices and all edge labels 3 and two short arms of consisting of single edges. The C­diagram of en is a linear diagram with n vertices and all edge labels 3 except for the last edge labelled 4. The C-diagram of D2 (k) is a single edge with label k. The C-diagrams of E 6 , E 7 , Es are star shaped with three arms and all edge labels 3. One arm has length one and another has length two. The C-diagram of F 4 is a linear diagram with edge labels 3,4, 3 in that order. The C-diagram of G 3 is a linear diagram with edge labels 3,5. The C-diagram of G 4 is a linear diagram with edge labels 3,3,5 in that order.

3. Elementary Quotient Operations

In this section we describe two types of elementary edge quotient opera­tions on a Coxeter system (W, S) of finite rank. The first we call edge label reduction and the second we call edge elimination.

Suppose sand t are distinct elements of S with m(s, t) < 00. Let d be a positive divisor of m = m(s, t), with d < m, and let N be the normal closure of the element (st)d of W. Then a presentation for WIN is obtained from the Coxeter presentation for (W, S) by adding the relator (st)d. As m = (mld)d, the relator (st)m is derivable from the relator (st)d and so the relator (st)m can be removed from the presentation for WIN.

Assume d > 1. Then the presentation for WIN is a Coxeter presentation whose P-diagram is obtained from the P-diagram for (W, S) by replacing the label m on the edge (s, t) with the label d. We call the operation of passing from the Coxeter system (w, S) to the quotient Coxeter system (WIN,{sN: s E S}) the (s,t) edge label reduction from m to d. For ex­ample, if we reduce the 4 edge of F 4 to 2, we obtain the Coxeter system A2 x A 2.

Now assume d = 1. We delete from the presentation for WIN the gen­erator t and the relator st and replace all occurrences of t in the remaining relators by s. Suppose r is in Sand k = mer, s) < 00 and e = mer, t) < 00.

Then we have the relators (rs)k and (rs)l in the presentation for WIN. Let d be the greatest common divisor of k and e. Then there are integers a and b such that d = ak+be. This implies that (rs)d is derivable from (rs)k and (rs)l and so we may add the relator (rs)d to the presentation for WIN. Then (rs)k and (rs)l are derivable from (rs)d and so we can eliminate the relators (rs)k and (rs)l from the presentation for WIN. We do this for each

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r in 8 such that m(r, s) < 00 and m(r, t) < 00. On the P-diagram level, we have eliminated the edge (s, t) and identified the vertices sand t and we have coalesced each edge (r, s) with label k < 00 with the edge (r, t) with label £ < 00 to form an edge with label d the greatest common divisor of k

and £. If each common divisor d is greater than one, we obtain a Coxeter presentation for WIN. If some common divisor d is one, we delete the cor­responding generator r and repeating the above reduction procedure on the presentation of WIN. As the set 8 of generators is finite, we will eventually stop deleting generators and obtain a Coxeter presentation for WIN with generators the subset 8' of {sN : s E 8} corresponding to the undeleted elements of 8. We call the operation of passing from the Coxeter system (W, 8) to the quotient Coxeter system (WIN, 8') the (s, t) edge elimination. For example, if we eliminate the 3 edge from C 3 , we obtain the Coxeter

system Al x AI'

4. The Binary Isomorphism Invariant

Let (W,8) be a Coxeter system of finite rank. For each pair of elements s, t of 8 with m(s, t) < 00, let b(s, t) be the 2-part of m(s, t), that is, b(s, t) is the largest power of 2 that divides m(s, t). If m(s, t) = 00, we define b(s, t) = 00. Let Nb be the normal closure in W of all the elements of the form (st)b(s,t) with b(s, t) < 00, and let Wb = WINb. Let TJ : W --> Wb be

the quotient homomorphism, and let 8b = TJ(8).

Proposition 4.1. The pair (Wb,8b) is a Coxeter system such that if s and t are in 8, then TJ(s) = TJ(t) if and only if sand t are conjugate in W. If sand tare nonconjugate elements of 8, then the order of TJ( S )TJ( t) is

the minimum of the set of all b( u, v) such that u and v are in 8 and u is conjugate to s and v is conjugate to t. In particular, the order of TJ(s)TJ(t) is a power of 2 or 00.

Proof. The system (Wb,8b) can be obtained from (W,8) by a sequence of elementary quotient operations. First we can do a series of edge label reductions of all the even labelled edges of the P-diagram of (W,8) to their 2-parts. Then we do a series of edge eliminations of all the odd la­belled edges. Each element of the form (st)b(s,t) with b(s, t) < 00 is in the

commutator subgroup of W. Therefore abelianizing W factors through the quotient WINb, and so TJ(s) = TJ(t) if and only if sand t are the same odd component of the P-diagram of (W, 8). Hence TJ(s) = TJ(t) if and only if s and t are conjugate in W by Prop. 3, p. 12 of Bourbaki [3].

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Suppose sand tare nonconjugate elements of Sand u and v are in S, with m( u, v) < 00, and u is conjugate to s and v is conjugate to t. Then u and v are not conjugate, and so m( u, v) is even, and therefore b( u, v) is a power of 2 greater than 1. In the coalescence of two such edges, the greatest common divisor is the minimum of the two edge labels. Therefore the order of TJ( s )TJ( t) is the minimum of the set of all b( u, v) such that u and v are in Sand u is conjugate to s and v is conjugate to t. 0

Theorem 4.1. Let (W, S) be a Coxeter system oj finite rank. For each pair oj elements s, t oj S with m(s, t) < 00, let b(s, t) be the largest power oj 2 that divides m(s, t). Let Nb be the normal closure in W oj all the elements oj the Jorm (st)b(s,t) with m(s, t) < 00, Then Nb is the normal closure in W

oj the set oj all elements oj W oj odd order. ThereJore Nb is a characteristic subgroup oj W that does not depend on the choice oj Coxeter generators S.

Proof. Every element of the form (st)b(s,t) with m(s, t) < 00 has odd order, and so Nb is contained in the normal closure of all the elements of odd order. Let w be an element of odd order, then TJ(w) has odd order in Wb = W/Nb. By the previous proposition, (Wb, Sb) is a Coxeter system with all edge labels a power of 2. Therefore TJ(w) is conjugate to an element of odd order of a finite visible subgroup of (Wb, Sb) by [3], Ch. V, §4, Ex. 2. The finite visible subgroups of (Wb, Sb) are direct products of groups of type Ai and C 2 , and so are 2-groups. Therefore Wb has no nontrivial elements of odd order. Hence TJ(w) = 1, and so w is in Nb. Thus Nb is the normal closure of all the elements of W of odd order. 0

P. Bahls proved in his Ph.D. thesis [1] that any finitely generated Cox­eter group has at most one P-diagram, up to isomorphism, with all edge labels even; see Theorem 5.2 in Bahls and Mihalik [2]. Therefore the iso­morphism type of the P-diagram of (Wb, Sb) is an isomorphism invariant of W by Theorem 4.2. We call the isomorphism type of the P-diagram of (Wb, Sb) the binary isomorphism invariant of W.

In Figure 1, we illustrate two P-diagrams and their binary isomorphism invariant P-diagrams below them. The even diagrams are not isomorphic, and so the top two P-diagrams represent nonisomorphic Coxeter groups.

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4 3 4 3 4 4

2

~2 4

~4 2

Figure 1

5. Matching Theorems

Let (W, 8) be a Coxeter system. A basic subset of 8 is a maximal irreducible subset B of 8 such that (B) is a noncyclic finite group. If B is a basic subset of 8, we call B a base of (W, 8) and (B) a basic subgroup of W. The theorems in this section are proved in our paper [8].

Theorem 5.1. (Basic Matching Theorem) Let W be a finitely generated Coxeter group with two sets of Coxeter generators 8 and 8'. Let B be a base of (W, 8). Then there is a unique irreducible subset B' of 8' such that [(B), (B)] is conjugate to [(B'), (B')] in W. Moreover,

(1) The set B' is a base of (W, 8'), and we say that Band B' match. (2) If I(B)I = I(B')I, then Band B' have the same type and there is an

isomorphism <p : (B) ----) (B') that restricts to conjugation on [(B), (B)] by an element of W.

(3) If I(B)I < I(B')I, then either B has type B2q+l and B' has type C 2q+l

for some q 2: 1 or B has type D 2(2q + 1) and B' has type D 2(4q + 2) for some q 2: 1. Moreover, there is a monomorphism <p : (B) ----) (B') that restricts to conjugation on [(B), (B)] by an element of W.

If a E 8, the neighborhood of a in P-diagram of (W, 8) is defined to be the set N(a) = {s E 8: m(s,a) < <Xl}.

If AS;;; 8, define A~ = {s E 8: m(s,a) = 2 for all a E A}.

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Theorem 5.2. Let (W, S) be a Coxeter system oj finite rank. Let B be a base oj (W, S) oj type C 2q+1 Jor some q ~ 1, and let a, b, c be the elements oj B such that m(a, b) = 4 and m(b, c) = 3. Then W has a set oj Coxeter

generators S' such that B matches a base B' oj (W, S') oj type B 2q+1 iJ and only iJ N(a) = B U B.l.

Theorem 5.3. Let B be a base oj (W, S) oj type C 2q+1 Jor some q ~ 1, and let a, b, c be the elements oj B such that m(a, b) = 4 and m(b, c) = 3.

Suppose that N(a) = B U B.l. Let d = aba, and let z be the longest element oj (B). Let S' = (S - {a}) U {d,z} and B' = (B - {a}) U {d}. Then S' is a set oj Coxeter generators Jor W such that

(1) The set B' is a base oj (W, S') oj type B 2q+1 that matches B, (2) (B').l = B.l U {z}, and {z} is a component oJ (B').l, (3) N(d) = N(z) = B' U (B').l,

(4) the basic subsets oj Sand S' are the same except Jor Band B'.

Theorem 5.4. Let (W, S) be a Coxeter system oj finite rank, and let B = {a, b} be a base oj (W, S) oj type D 2 ( 4q + 2) Jor some q ~ 1. Then W has

a set oj Coxeter generators S' such that B matches a base B' oj (W, S') oj

type D 2 (2q + 1) iJ and only iJ either v = a or v = b has the property that N(v)=BUBJ...

Theorem 5.5. Let B = {a, b} be a base oj (W, S) oj type D 2 ( 4q + 2) Jor some q ~ 1. Suppose that N(a) = B U B.l. Let c = aba and let z be the longest element oj (B). Let S' = (S - {a}) U {c, z} and B' = {b, c}. Then S' is a set oj Coxeter generators oj W such that

(1) The set B' is a base oj (W, S') oj type D2(2q + 1) that matches B, (2) (B').l = BJ.. U {z}, and {z} is a component oJ (B').l,

(3) N(c) = N(z) = B' U (B').l, (4) the basic subsets oj Sand S' are the same except Jor Band B'.

6. The Even Isomorphism Invariant

Let (W, S) be a Coxeter system of finite rank. A pair of elements (a, b) of S is said to be unreduced if (a, b) satisfy the conditions of Theorem 5.4, that is, m( a, b) == 2 mod 4 and either v = a or v = b has the property that N(v) = {a,b} U {a,b}.l; otherwise the pair (a, b) is said to be reduced.

Let (a, b) be a pair of elements of S. If (a, b) is unreduced, define m(a, b) = 2, and if (a, b) is reduced, define m(a, b) = m(a, b).

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Let Ne be the normal closure in W of all the elements of the form ab with a and b elements of B such that m(a, b) is odd together with all the elements of the form (ab)2 such that (a, b) is an unreduced pair of elements of B. Let We = WINe, let 'r/ : W ---7 We be the quotient homomorphism,

and let Be = 'r/(B).

Proposition 6.1. The pair (We, Be) is a Coxeter system such that if sand t are in B, then 'r/(s) = 'r/(t) if and only if sand t are conjugate in W. If sand tare nonconjugate elements of B, then the order of'r/(s)'r/(t) is the greatest common divisor of the set of all m( u, v) such that u and v are in Band u is conjugate to s and v is conjugate to t. In particular, the order

of'r/(s)'r/(t) is either even or 00.

Proof. The system (We, Be) can be obtained from (W, B) by a sequence of elementary quotient operations. First we reduce to 2 all the edge labels of unreduced edges (a, b) to obtain a Coxeter system with Coxeter matrix M = (m(s, t))8,tES. Then we eliminate all the odd labelled edges. Each element of the form either st, with m(s,t) odd, or (st)2, with (s,t) unreduced, is in the commutator subgroup of W. Therefore abelianizing W factors through the quotient WINe, and so 'r/(s) = 'r/(t) if and only if sand t are in the same odd component of the P-diagram of (W, B). Hence 'r/(s) = 'r/(t) if and only if sand t are conjugate in W by Prop. 3, p. 12 of Bourbaki [3J.

Suppose sand t are nonconjugate elements of Band u and v are in B, with m(u,v) < 00, and u is conjugate to s and v is conjugate to t. Then u and v are not conjugate, and so m( u, v) is even. In the coalescence of two such edges, the greatest common divisor is even. Therefore the order of 'r/(s)'r/(t) is the greatest common divisor of the set of all m(u,v) such that u and v are in Band u is conjugate to s and v is conjugate to t. D

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Let (W, S) be a Coxeter system of finite rank. A base B of (W, S) is said to be of odd type if there are elements a and b in B, with m( a, b) odd. A base B of (W, S) is said to be reduced if for every set of Coxeter generators S' of W, the base B matches a base B' of (W,S') with I(B)I:::; I(B')I. Note that a base B of (W, S) is unreduced precisely when B satisfies the conditions of either Theorem 5.2 or 5.4. In particular, a base B = {a, b} of rank 2 is unreduced if and only if the pair (a, b) is unreduced.

Theorem 6.1. Let (w, S) be a Coxeter system of finite rank. Then Ne is a characteristic subgroup of W that does not depend on the choice of Coxeter generators s.

Proof. Observe that Ne is the normal closure in W of the commutator subgroups of all the basic subgroups (B) of (W, S) such that the base B is either of odd type or unreduced of rank 2. Let S' be another set of Coxeter generators of W. By the Basic Matching Theorem and Theorem 5.4, the group Ne is also the normal closure in W of the commutator subgroups of all the basic subgroups (B') of (W, S') such that the base B' is either of odd type or unreduced of rank 2. Therefore Ne is the normal closure in W of all the elements of the form s't' with s' and t' in S' and m(s', t') odd together with all the elements of the form (S't,)2 with (s', t') an unreduced pair of elements of S'. Thus Ne is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S. D

Let (W, S) be a Coxeter system of finite rank. P. Bahls proved in his Ph.D. thesis [1] that any finitely generated Coxeter group has at most one P-diagram, up to isomorphism, with all edge labels even. Therefore the isomorphism type of the P-diagram of (We, Se) is an isomorphism invariant of W by Theorem 6.2. We call the isomorphism type of the P-diagram of (We, Se) the even isomorphism invariant of W. For example, the even isomorphism invariant of the system D2(6) is the isomorphism type of the P-diagram of the system Al x A 1 .

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7. Basic Characteristic Subgroups

Let (W, S) be a Coxeter system of finite rank, and let F be a family of finite irreducible Coxeter system isomorphism types. Let N(F) be the nor­mal closure in W of the commutator subgroups of all the reduced basic subgroups of (W, S) of isomorphism type contained in F together with the commutator subgroups of all the unreduced basic subgroups of (W, S) that match a reduced basic subgroup of another system (W, S') of isomorphism type contained in F. Let W(F) = W/N(F) , let ry : W ---) W(F) be the quotient homomorphism, and let S(F) = ry(S).

Theorem 7.1. The pair (W(F), S(F)) is a Coxeter system that can be ob­tained from (W, S) be a finite series of elementary edge quotient operations. The group N(F) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S.

Proof. Quotienting out the commutator subgroup of a basic subgroup (B) of (W, S) can be realized by reducing all the even labelled edges of the C­diagram of ((B), B) to 2 and eliminating all the odd labelled edges of the C­diagram. Therefore (W(F), S(F)) is a Coxeter system that can be obtained from (W, S) be a finite series of elementary edge quotient operations.

Let S' be another set of Coxeter generators of W. By the Basic Matching Theorem, N(F) is also the normal closure in W of the commutator sub­groups of all the reduced basic subgroups of (W, S') of isomorphism type contained in F together with the commutator subgroups of all the unre­duced basic subgroups of (W, S) that match a reduced basic subgroup of isomorphism type contained in F. Thus N(F) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S. 0

We call a subgroup of W of the form N(F) a basic characteristic sub­group.

Corollary 7.1. If W is a finitely generated Coxeter group and N(F) is a basic characteristic subgroup of W, then the isomorphism type of W(F) =

W/N(F) is an isomorphism invariant of W.

Note if F is the family of all finite irreducible Coxeter system isomor­phism types, then N(F) is the commutator subgroup of W, and W(F) is the abelianization of W.

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8. The Spherical Rank Two Invariant

B. Miihlherr [7] has announced a solution of the isomorphism problem for finitely generated Coxeter groups W such that W has no basic subgroups of rank greater than 2 with respect to some set of Coxeter generators. By the Basic Matching Theorem, if W has no basic subgroups of rank greater than 2 with respect to some set of Coxeter generators, then W has no basic subgroups of rank greater than 2 with respect to every set of Coxeter generators. Therefore it makes sense to say that W has no basic subgroups of rank greater than 2 without regard to a set of Coxeter generators.

In this section we describe a characteristic subgroup N2 of a finitely generated Coxeter group W such that W2 = W/N2 is a Coxeter group with no basic subgroups of rank greater than 2 and such that the isomorphism type of W 2 is an isomorphism invariant of W.

Let Xn be one of the finite irreducible Coxeter systems An, Bn, Cn, E 6 ,

E 7, Es, F 4, G 3 , G4 of rank n 2: 3. We now define a characteristic subgroup N(Xn) of Xn for each X n . Let N(Xn) be the commutator subgroup of the Coxeter group Xn if n 2: 5 or if Xn = A 4 , G 3 , or G 4 •

Let al, a2, a3 be the Coxeter generators of A3 indexed so that m(al, a2) = m(a2, a3) = 3. Let N(A3) be the normal closure in the group A3 of the element ala3' Then N(A3) is a characteristic subgroup of A3

characterized by the property that N(A3) is the unique normal subgroup of A3 such that A3/N(A3) is isomorphic to A2 according to Table 3 of

Maxwell [6]. Let bl , b2 , b3 , b4 be the Coxeter generators of B4 indexed so that

Let N(B4) be the normal closure in the group B4 of the elements bl b2

and b2b3 . Then N(B4) is a characteristic subgroup of B4 characterized by the property that N(B4) is the unique normal subgroup of B4 such that B4/N(B4) is isomorphic to A2 according to Table 3 of Maxwell [6].

Let Cl, C2, C3 be the Coxeter generators of C 3 such that m(cl, C2) = 3 and m( C2, C3) = 4. Let N (C3 ) be the normal closure in the group C 3

of the element (C2C3)2. Then N(C3) is a characteristic subgroup of C 3

characterized by the property that N(C3) is the unique normal subgroup of C 3 such that C3/N(C3) is isomorphic to A2 x Al according to Table 3

of Maxwell [6]. Let Cl, C2, C3, C4 be the Coxeter generators of C 4 indexed so that

224

Let N(C4 ) be the normal closure in the group C4 of the element CIC3. Then N(C4 ) is a characteristic subgroup of C 4 characterized by the property that N(C4) is the unique normal subgroup of C 4 such that C 4/N(C4) is isomorphic to A2 x Al according to Table 3 of Maxwell [6].

Let fl, 12,13, f4 be the Coxeter generators of F 4 indexed so that

m(fl, h) = m(h, f4) = 3 and m(h, h) = 4.

Let N(F4) be the normal closure in the group F4 of the element (1213)2. Then N (F 4) is a characteristic subgroup of F 4 characterized by the property that N(F4) is the unique normal subgroup of F4 such that F4/N(F4) is isomorphic to A2 x A2 according to Table 3 of Maxwell [6].

Let (W, S) be a Coxeter system of finite rank. Let N(W) be the normal closure in W of the subgroups N( (B)) defined above for every base B of (W, S) of rank greater than 2. Let W(2) = W/N(W). Let ry : W --4 W(2) be the quotient homomorphism, and let S(2) = ry(S).

Theorem 8.1. The pair (W(2), S(2») is a Coxeter system that can be ob­

tained from (W, S) be a finite series of elementary edge quotient operations. The group N(W) is a characteristic subgroup of W that does not depend on the choice of Coxeter generators S.

Proof. Quotienting out the group N( (B)) for each base B of (W, S) of rank greater than 2 can be realized by elementary edge quotient operations. Therefore (W(2), S(2») is a Coxeter system that can be obtained from (W, S)

be a finite series of elementary edge quotient operations. Let S' be another set of Coxeter generators of W. By the Basic Match­

ing Theorem and the characteristic properties of the groups N(Xn), the group N(W) defined in terms of the generators S is the same as the group N(W) defined in terms of the generators S'. Thus N(F) is a characteristic subgroup of W that does not depend on the choice of generators S. D

Corollary 8.1. If W is a finitely generated Coxeter group, then the iso­morphism type of W(2) = W/N(W) is an isomorphism invariant of W.

It may happen that (W(2), S(2») has a base of rank greater than 2. To get a quotient system with no bases of rank greater than 2, we may have to quotient out N(W(2»), and then perhaps repeat the above quotienting process several times. This leads to a finite nested sequence

225

of characteristic subgroups of W such that if WCi) = W/N(i)(W) and if 'rJi : W ----> WCi) is the quotient homomorphism, then

N Ci+1)(W) = 'rJ;-l(N(WCi)))

for each i = 1, ... , £ - 1, and WCe) has no basic subgroups of rank greater than 2, and £ is as small as possible. We have that WCHl) = (WCi))C2) for each i = 1, ... , £ - 1. Therefore the isomorphism type of W(i) for each i = 1, ... , £ is an isomorphism invariant of W. It follows from the Basic Matching Theorem that £ does not depend on a choice of Coxeter generators for W, and so £ is an isomorphism invariant of W. We call £ the spherical rank 2 class of W. We have £ ~ 1 with £ = 1 if and only if W has no bask subgroups of rank greater than 2. Figure 2 shows the P-diagrams of a sequence WCl), ... , WCl) with £ = 4 for the Coxeter group W = WCl).

Define N2 = N(£)(W). Then N2 is a characteristic subgroup of W such that W2 = W / N2 has no basic subgroups of rank greater than 2. The isomorphism type of W2 is an isomorphism invariant of W which we call the spherical rank 2 isomorphism invariant of W.

Let 'rJ : W ----> W2 be the quotient homomorphism, and let S2 = 'rJ(S). Then (W2' S2) is a Coxeter system that can be obtained from (W, S) by a finite series of elementary edge quotient operations.

2

3 2

3 3

3 2

3 3

Figure 2

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9. Conclusion

Let (W,8) be a Coxeter system of finite rank. In this paper, we have de­scribed three characteristic subgroups Nb, N e, N2 of W each leading to a quotient isomorphism invariant of W. It is interesting to note that

N2 ~ Ne ~ Nb,

and so the quotient isomorphism invariants corresponding to Nb, N e, N2 are progressively stronger. The algorithm for finding a P-diagram for the system (Wb, 8b) starting from a P-diagram of (W, 8) is computational fast.

The algorithm for finding a P-diagram for the system (We, Se) is slower since it has to determine the bases of (W, 8) of type D 2 (4q + 2) that satisfy the conditions of Theorem 5.4; but, this algorithm is only slightly slower since the conditions of Theorem 5.4 are easy to check. The algorithm for finding a P-diagram for the system (We, Se) would most likely be incor­porated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups, since the even isomorphism invariant would usually determine that two random finite rank Coxeter systems have nonisomorphic groups.

The algorithm for finding a P-diagram for the system (W2,82) is the slowest, but it is not much slower, since it only has to find a subdiagram of the P-diagram of (W, 8) of type A 3 , C 3 or G 3 before it performs an edge quotient operation on an edge of the subdiagram, and therefore reduces the complexity of the P-diagram. If the sub diagram is of type A3 or G 3 , then the edge with label 2 is eliminated. If the subdiagram is of type C 3 , then the 4 edge label is reduced to 2. The algorithm then repeats the routine of searching for a subdiagram of type A 3 , C 3 or G 3 and performing the corresponding edge quotient operation.

The algorithm for finding a P-diagram for the system (W2,82 ) would most likely be useful in an efficient program that determines if two finite rank Coxeter systems have isomorphic groups, since the solution of the isomorphism problem for finite rank Coxeter systems that have no bases of rank greater than 2 is considerably simpler than any general solution of the isomorphism problem.

227

References

l. P. Bahls, Even rigidity in Coxeter Groups, Ph.D. Thesis, Vanderbilt Univer­sity,2002.

2. P. Bahls and M. Mihalik, Reflection independence in even Coxeter groups, Geometriae Dedicata 110 (2005), 63-80.

3. N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5, et 6, Hermann, Paris, 1968.

4. H.S.M. Coxeter, The complete enumeration of finite groups of the form R; = (RiRj)kii = 1, J. London Math. Soc. 10 (1935), 21-25.

5. H.S.M. Coxeter, Regular Polytopes, Dover, New York, 1973. 6. G. Maxwell, The normal subgroups of finite and affine Coxeter groups, Proc.

London Math. Soc. 76 (1998), 359-382. 7. B. Miihlherr, The isomorphism problem for Coxeter groups, In: The Coxeter

Legacy: Reflections and Projections, Edited by C. Davis and E.W. Ellers, Amer. Math. Soc., (2006), 1-15.

8. M. Mihalik, J. Ratcliffe, and S. Tschantz, Matching theorems for systems of finitely generated Coxeter groups, Algebr. Geom. Topol. 7 (2007), 919-956.

Localization and I A-automorphisms of finitely generated, metabelian, and torsion-free nilpotent groups

Marcos Zyman *

Department of Mathematics, The City University of New York-BMCC New York, New York 10007, USA E-mail: mzyman@bmcc.cuny.edu

Given a nilpotent group G and a prime p, there is a unique p-local group G(p) which is, in some sense, the "best approximation" to G among all p-local nilpotent groups. G(p) is called the p-localization of G. Let I A( G) be the group of automorphisms of G that induce the identity on G/[G, G]. IA(G) turns out to be nilpotent so its p-localization exists. Two groups are said to be in the same localization genus if their p-localizations are isomorphic for all p. We prove that if two finitely generated, torsion-free nilpotent, and metabelian groups lie in the same localization genus, their I A-groups also lie in the same localization genus. The method of proof involves basic sequences and commutator calculus.

Keywords: Nilpotent groups, p-localization, I A-automorphisms.

1. Introduction

The objective of this paper is to investigate the interaction between the I A­automorphisms and p-localization of finitely generated, torsion-free nilpo­tent, and metabelian groups. In particular we prove that if two such groups lie in the same localization genus, their I A-groups also lie in the same local­ization genus. The proof requires an understanding of basic commutators and commutator calculus. Before stating the main theorem precisely, we discuss some initial notions and facts (see Refs. 1, 2, and 3).

A group G is called p-local if the map x f--+ xn from G to itself is a bijection for n relatively prime to p. For every nilpotent group G there is a homorphism of nilpotent groups e : G -4 G(p) which p-localizes G . This means that G(p) is p-local, and for every p-local nilpotent group K, the map e* : Hom (G(p),K) -4 Hom(G,K) given by e*(<p) = <pe is a bijection.

'This paper is based on the author's doctoral dissertation, written under the guidance of Joseph Roitberg at the City University of New York.

228

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Observe that Z(p) = {min: m,n E Z and (n,p) = I} is the p-localization of the additive group of integers, and e : Z ----? Z(p) is the obvious embedding.

The nilpotency class of G(p) never exceeds that of G and p-localization is a functor from the category of nilpotent groups to itself. Furthermore, the restriction of e to each term of the lower central series of G also gives a p-localization map. Finally, two nilpotent groups are said to be in the same localization genus if their p-localizations are isomorphic for every prime p.

The group of I A-automorphisms of G is the subgroup of AutG consist­ing of those automorphisms that induce the identity on GIG', where G' is the commutator subgroup of G.

We are now in a position to state the main theorem:

Theorem 1.1. Let G and H be finitely generated, metabelian, and torsion­free nilpotent groups that lie in the same localization genus. Then I A( G) and I A( H) are finitely generated and torsion-free nilpotent groups, which also lie in the same localization genus.

Theorem 1.1 can be applied to the following examples, which we discuss further in §4. In the category of class 4 nilpotent groups consider Remeslen­nikov's groups

Fs = (x, y; [y, x, y, y]3[y, x, x, y][y, x, x, X]2),

and

Fr = (x,y; [y,x,y,y]6[y,x,x,y][y,x,x,xj).

It is the case that Fs and Fr are non-isomorphic but they lie in the same localization genus. These groups satisfy the conditions of Theorem 1.1, so their I A-groups also lie in the same localization genus. Providing an additional source of examples, Pickel and Roitberg (see Ref. 4) describe a family of non-isomorphic class 4 nilpotent groups that also satisfy the hypothesis of Theorem 1.1 and prove that they lie in the same localization genus. Again, our theorem can be applied to their I A-automorphisms.

We also argue in §4 that Theorem 1.1 is non-trivial by carrying out explicit computations of the inner automorphisms of G, as well as of I A( G) where G is free nilpotent of class 2 and rank 3. These calculations give that I nnG =1= I A( G) in general.

I am indebted to Joseph Roitberg, my advisor, who suggested the prob­lem and guided me through the research process; as well as to Gilbert

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Baumslag for all his help and support. I also wish to thank Marianna Bo­nanome, Margaret H. Dean, and Katalin Bencsath for many helpful con­versations. During the early stages of my research I used the group theory package MAGNUS (available at http://sourceforge.net/projects/magnus), developed by the Center for Algorithms and Scientific Software at the City College of New York.

This paper is dedicated to Joseph Roitberg.

2. Preliminary discussion

For any group G let "fiG denote the i-th term of the lower central series, where "f1 = G and "f2G is the commutator subgroup of G. The following result of P. Hall (see Ref. 5) is the departing point of our discussion.

Theorem 2.1. Let G be nilpotent of class c. Then

• "fjIA(G) transforms each "fiGhi+jG identically, and • I A( G) is nilpotent of class c - 1.

The fact that G modulo its center is isomorphic to the group of inner automorphisms of G also suggests that IA(G) may be torsion-free if Gis torsion-free. In fact, the following is true:

Lemma 2.1. If G is a finitely generated, torsion-free nilpotent group, I A( G) is also finitely generated, and torsion-free nilpotent.

Lemma 2.1 can be proved by induction on the class of G (see Lemmas (1.2.9) and (1.2.10) in Ref. 6).

Assuming nilpotency of G is crucial here. C.K. Gupta proved that I A( G) is metabelian if G is a two generator metabelian group (see [7J in Ref. 7). However, IA(G) need not be finitely generated (see Theorem C in Ref. 7).

As a corollary of Theorem 2.1 and Lemma 2.1, we have:

Corollary 2.1. If G is finitely generated and torsion-free nilpotent of class c, then I A( G) is finitely generated and torsion-free nilpotent of class c - 1.

The following definitions and remarks (see Ref. 2) lead to a very useful reformulation of Theorem 1.1.

Let <p : G ~ H be a homomorphism of torsion-free nilpotent groups. <p is called a p-isomorphism if (i) <p is injective, and (ii) for every h E H

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there is an n, relative prime to p, such that hn belongs to the image of cpo A homomorphism satisfying (ii) is called p-surjective .

The proof of Theorem 1.1 depends on the fact that a homomorphism of nilpotent groups cp : G ---; H is a p-localization map if, and only if:

(1) His p-local, and (2) cp is a p-isomorphism.

In fact, we actually prove the following statement, from which Theorem 1.1 readily follows:

Theorem 2.2. Let G be finitely generated, metabelian, and torsion-free nilpotent. Then I A( G(p)) is p-local, and the natural map

is a p-isomorphism.

The motivation for Theorem 2.2 comes from a related result by Maruyama (see Ref. 8) in homotopy theory. To describe it, let X be a simply connected CW-complex and denote by co(X) the group of homo­topy classes of self-homotopy equivalences of X that induce the identity on all homology groups. E. Dror and A. Zabrodsky proved that co(X) is nilpotent (see Ref. 9), so its p-localization makes sense. Maruyama's re­sult is that the homomorphism co(X) ---; co(X(p)) obtained by localizing each homotopy class is in fact the localization homomorphism of nilpotent groups co(X) ---; co(X)(p). Theorem 2.2 does not follow from Maruyama's result since X is assumed to be simply connected, and the elements of co(X) induce the identity an all homology groups.

3. Proof of the main theorem

3.1. Powers of IA-automorphisms

We first develop a technique to compute powers of I A-automorphisms. Let G be the p-localization of a finitely generated, torsion-free, metabelian, and nilpotent group of class c.

Let X = {Xl, ... ,xr } be a finite set that generates G as a p-local group. Then G is also generated, as a p-local group, by the set of basic commu­tators B = {bl , ... , bm } on X; where bi = Xi for i = 1, ... , r. Since G is metabelian, any basic commutator on B is of the form

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Denote the weight of bi by wt(bi ).

Let <p E I A( G) and put

where wt(bi ) ::::; C - 1 and Ai E G'. Ai can be expressed as a product of rational powers of basic commutators of weight at least 2, and at most c.

So we can write

Ai = II [bk,Xz]* k>l

where wt(bk) ::::; C - 1 for each k; and v(i) is relatively prime to p. Direct computations involving commutator calculus give

where

Dil = II([bk,Ad[Ak,xz])*&. k>l

In general, for each i, we may construct a sequence of elements of G':

where

Dil = II([bk, Az][Ak,xz])*, k>l

Dij = II([h,Dl(j-l)][Dk(j-I),xd)~ for j > 1, k>l

<p(Ai) = AiDil, and <p(Dij) = DijDi(J+I)' Notice that if Di(j-I) E 1ZG for some integer z, then Dij E Iz+jG. We refer to Sequence 1 as the D-sequence associated to Ai. A straightforward induction on m now gives the following lemma:

Lemma 3.1.

(()m(A.) _ A. ,Cl(m) ,C2(m) rc",(m) ..- t - tUil Ui2 •.. Uim

where

(m) m! Cj(m) = , = "( _ ')"

J J. m J,

(1)

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Further computation and Lemma 3.1 give:

Lemma 3.2.

rpm(b) = A.rp(k)rp2(k) ... rpm-l(k) = b-Am8dl(m)8d2(m) ...• dm - 1 (m) t t t t t t t tl t2 U i( m-l) ,

where

3.2. Proof that I A( G) is p-local if G is p-local

Consider the map

IA(G) ~ IA(G)

rp f--+ rpn

where (n,p) = 1, and G is the p-localization of a finitely generated, metabelian, and torsion-free nilpotent group of class c. We now embark on the proof that this map is a bijection.

Lemma 3.3.

IA(G) ~ IA(G)

is one-to-one.

Proof. To see this, let

rpn = '1f;n.

We wish to prove that rp = '1f;. For this purpose put

rp(bi ) = biAi,

where 1 :::; wt(b i ) :::; C - l.

As usual we write

and

(2)

(3)

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In order to show that cp = 'Ij; we proceed by reverse induction on wt(bi ). Suppose wt(bi ) = c - 1. Then both Ai and ..t belong to IC' cpn(bi ) =

'lj;n(bi ) implies that biAi = buii, so that Ai = Ai. By p-locality this means

that Ai = Ai. Now suppose that bi satisfies

Assume the induction hypothesis that cp = 'Ij; on 1}+1. Our goal is to show that cp(bi ) = 'Ij;(bi ). Since cpn(bi ) = 'lj;n(bi ) then

b.An8d,(n) ... 8dn-,(n) = b.AnJdl(n) ... Jdn-l(n) , , >1 i(n-l) ',>1 i(n-l) .

Using the fact that we are in a p-local group, we obtain:

A_I -1 A dl(n) -1 A dn_l(n) AiAi = (8i1 8il ) n ... (8i(n_l)8i(n-l)) n. (4)

Since wt(bi ) = j, Ai and Ai each belongs to Ij+!' In fact, by Eq. 4, the product AiAi1 actually lies in 1}+2. By induction:

cp(Ai) = 'Ij;(Ai) = AiJi1 .

This means that A_I A A -1 A_I A_I

cp(AiAi ) = Ai8il(Ai8il) = AiAi 8i1 8i1 .

Simply because cp is and I A-automorphism, it follows that A_I

8i1 8i1 E 1}+3.

Similarly, if we evaluate cp on any 8imJ~ in the right hand side of Eq. 4 we have (again by induction on j) that

(5)

since

Again, because cp is and I A-automorphism, Eq. 5 implies that 8i (m+!) Ji(;;'+I) is in a higher commutator subgroup than the commuta-

tor subgroup where 8imJ~ lies. Hence the entire right-hand side of Eq. 4 belongs to 1}+3. We have established that

A_I AiAi E 1}+3.

Apply the same argument repeatedly to finally conclude that A_I

AiAi E Ie+! = 1.

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This completes the proof that I A( G) -+ I A( G) is one-to-one. o

Lemma 3.4. Let G be the p-localization of a finitely generated, metabelian, and torsion-free nilpotent group of class c, and let n be relatively prime to p. Then the map from I A( G) to itself given by cp I-' cpn is onto.

Proof. Let {b1 , b2 , ... } be the basic commutators on X = {Xl, •.. , xr } of weight at most c - 1. Let cp(bi) = biAi E I A( G).

Let Oil, Oi2, ... ,Oi(c-2) be the o-sequence associated to Ai' (Since G has class c, we can henceforth assume that Oil = 1 for I > c - 2). We claim that there are p-local integers E 1 (n), ... , Ec- 2 (n) that depend on n (and c), such that

• ·'·(b) = b·A;\:o<l(n) .. ·0<C-2(n) E IA(G) and 'f' 2 2 2 d 2( c-2) ,

• 1/Jn = cpo

Put

Ai = II[bk, Xzlikl

,

k>l

where the ikl are p-local integers. We wish to find p-local integers E1, ... ,Ec-2, depending on nand c alone,

such that

1

gives an I A-automorphism where Qi = A; oa ... 0:(;~2) and 1/Jn = cpo We will show that these E'S can in fact be found by "solving" the equation 1/Jn = cpo

Associated with Qi we have the corresponding sequence of deltas:

Ji1 , ... , Ji (c-2)'

Commutator calculus gives: 1

o"(b·) = b -A n oEt ........... ·0<c-2 ) 'f' 2 2 2 d 2(c-2

1

.I'(A-) = A-on 0<1 ......... 0«C-3 ) 'f' 2 2 d 22 2 c-2

1

1/J(Oi1) = Oi10i~0:j ...... O:(;~2) 1

1/J( Oi2) = Oi20Z~ 0:4 ... 0:(;~2)

Additional calculations yield:

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151/ n2 15"'1 ... 15"'c-3 and aJ' depends on

il i2 i(c-2)'

n,EI,···,Ej.

• 'IjJ(Jid = JiI Ji2 where Ji2 151/n315i31 .. . 15i3c - 4 and f3J' depends on

i2 i3 i(c-2)'

We continue to find similar expressions for Jij in terms of the 15ij , the last expression being

A (l/n)c-1 15i (c-2) = 15i (c-2)

We now show that the E'S appearing in the equation

(6)

can be chosen so that they depend only on n. For this purpose we use the formula that describes powers of IA-automorphisms (see Lemma 3.2). Equation 6 becomes:

A15n€1 ... 15nfc- 2 Jd1 ... Jd(c-2) = A t tl t(c-2) tl t(c-2) t·

Canceling Ai and rewriting the J's in terms of the 15's yields:

8nE1 , .. 8n €c-2 8~8"'1 , .. 8"'c-3 8-;;J8f31 ... 8f3c-4 , .. 8 nc - 1 -1 ( 1 )d1( 1 )d2 ( 1 )dC_2

i1 i(c-2) i1 i2 i(c-2) i2 i3 i(c-2) i(c-2) -,

Now solve for each Ei in the following way:

• nEI + ,&dl = 0 =} EI = - ~dl' which means that EI can be chosen so that it depends on n only.

• nE2 + aIdl + Ad2 = 0 =} E2 = -laIdl - Ad2. Since al depends on n n n n

and EI, we conclude that E2 can be chosen so that it depends on n only.

• nE3 + a2dI + f3l d2 + Ad3 = 0 =} E3 = -la2dl - lf3l d2 - Ad3. Again, n n n n

a2 depends on n, EI, and E2; so E3 can be chosen in such a way that it depends on n alone.

Continue this process to choose EI, ... , Ec-2 so that they depend only on n. Computations show that 'IjJ lifts to a well-defined I A-automorphism (see the second part of the proof of theorem (2.2.2) in Ref. 6). Lemma 3.4 is ~~~. 0

3.3. IA(G) -+ IA(G(p» is a p-isomorphism

Let G = gp(XI, . .. , xm) be a finitely generated, metabelian, and torsion­free nilpotent group of class c. Let G(p) be its localization at the prime p.

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Consider the localization diagram

G ~ G

fp -----7 G(p)

and observe that if! E IA(G), then !p E IA(G(p»).

Lemma 3.5. The homomorphism

is a monomorphism.

Proof. Let! E ker (IA(G) -+ IA(G(p»)) and put !(Xi) = XiAi where Ai lies in G' = [G, G]. For 9 E G, write e(g) = g. If Xi is a generator of G, Xi belongs to G(p) and

Xi = !P(Xi) = !(Xi) = xiAi.

Hence, Ai = I in G~p), which means that Ai belongs to ker (e : G' -+ G~P»)' Since G' is torsion-free and e : G' -+ G~p) is a localization map (see §1),

e is one-to-one so that Ai = 1. Hence !(Xi) = Xi; and therefore

is indeed a monomorphism. o

Remark. The discussion that led to the b-sequence (see 1 in § 3) and Lemma 3.2 remains valid for I A( G), where G is finitely generated, torsion­free nilpotent, and met abelian (the condition of G being p-local is dropped). This observation will be used in the sequel.

Lemma 3.6. IA(G) -+ IA(G(p») is p-surjective.

Proof. Let <p E IA(G(p»). Consider the action of <p on the "p-generators" of G(p): <P(Xi) = XiAi, where Ai E G~p)" Since e : G' -+ G~p) is p-surjective,

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there exists an integer Si, relatively prime to p, such that A:i belongs to the image of e : G' -. G~p) for each i = 1,2, ... , m. Put

A~l clearly lies in the image of e : G' -. G~p) (for each i = 1,2, ... , m)

because such image is a subgroup of G~p)' Similarly, choose (}2,···, ()c-l

(independent if i) so that O~:+l belongs to the image of e : rk+2(G) -. rk+2(G(p)), for k = 1, ... , c - 2. Let

We have the following:

(1) () is relatively prime to p,

(2) Ai belongs to the image of e : G' -. G~p), and (3) Oik belongs to the image ofe : rk+2(G) -. rk+2(G(p)), (k = 1, ... ,c-2).

Using Lemma 3.2, we see that

d ( ) - (a) _ a(a-l) • 1()-2- 2

d ( ) - (a) _ a(a-1)(a-2) • 2 () - 3 - 3!

d ( ) - (a) _ a(a-1)(a-2)(a-3) • 3 () - 4 - 4! . .

d () - ( a ) _ a(a-1)(a-2) ... (a-c+2) • c-2 () - c-1 - (c-1)! .

Fix 1 :::; j :::; c - 2 and consider the number

. _ ( () ) _ ()(()-1)(()-2) .. ·(()-j) dJ (()) - j + 1 - (j + I)! .

Write

where p and Ej are relatively prime. (If p does not divide (j + I)! take (Xj = 0.) Next, let

Notice that p and E are relatively prime. Now put

S = E().

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Again, p and s are relatively prime. Invoking Lemma 3.2 once more, we see that for each j:

dj(s) = f1 f 2··· f c_2 U (s-1)(s-2)···(s-j) = pOiJfj

f1f2··· fj-1fJ+1 ... fC-2U(S - l)(s - 2)··· (s - j) pOij

As dj(s) is an integer, and pOij does not divide f1f2··· fj-1fJ+1··· fc-2U,

pOij has to divide (s - l)(s - 2)··· (s - j). Hence, each integer s,d1(s),d2(s), ... ,dc- 2 (s) is a multiple of u. The crucial conclusion is that

(1) s is relatively prime to p,

(2) Ai belongs to the image of e : G' --> G~p), and

(3) 151;(8) lies in the image of e : I'J+2(G) --> I'J+2(G(p)), for j = 1,2, ... , c - 2.

We can therefore choose C¥i E G' such that ai = Ai, and Dij E I'J+2 (G)

such that Dij = 151/8) for each j = 1,2, ... , c- 2; where Ii = e(g) for 9 E G. Using Lemma 3.2 yet again we see that

8 (-.) _ - -A8 r5d1 (8) ... r5dC-2 (8) cp X. - x. • i1 i(c-2) .

Let

fJi = C¥iDi1 ... Di(c-2) E G'.

Define the following map on the generators of G:

f(Xi) = xifJi·

It is routine to show f can be extended to an element of I A( G) (see p. 47 of Ref. 6). Then, fp and cp8 coincide on the p-generators of G(p) and will therefore be equal as I A-automorphisms. 0

The proof of Theorem 2.2 is now complete, and Theorem 1.1 readily

follows.

4. Examples

4.1. An example where InnG =I IA(G)

Let (denote the center of G. If it were the case that InnG = IA(G) for all nilpotent groups G, Theorem 1.1 would be trivial since we would have

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due to the facts that (G/()(p) ~ G(p)/(p) (see Ref. 2) and H/((H) ~ InnH for any group H.

To demonstrate that Theorem 1.1 is nontrivial in general we compute I nnG and I A( G) where G is free nilpotent of class 2 and rank 3. It will then be clear that InnG -=f. IA(G).

Let

G=(x,y,z)

be free nilpotent of class 2 on the generators {x,y,z}. Put C12 = [x,y], C13 = [x,z], and C23 = [y,z] (we will use similar notations in §4.2). Then

is a basic sequence of basic commutators on {x, y, z}. Since G is free nilpo­tent, every g E G can be uniquely written as

I , I e' e' e' If g' = xelye2ze3c1~2c133c233 is another element of G, standard commutator calculus gives

,_ el+e~ e2+e; e3+e; e12+e~2-e;e2 e13+e;3-e;ea e23+e~3-e~e3 gg - X Y Z C12 C13 c23 . (7)

Consider the following nine elements of I A( G):

<t?l(X) = XC12, <t?l(Y) = y, <t?l(Z) = z; <t?2(X) = x, <t?2(y) = YC12, <t?2(Z) = z;

<t?g(x) = x, <t?g(y) = y, <t?g(z) = ZC23·

For any <t? E I A( G) we can write

( ) az bz Cz <t? z = zC12C13c23·

It is straightforward to show that <t? is uniquely expressed as

In - Inax ",ay Inaz Inbx ,nby ,nbz InCX Incy InCZ ..- -..-1 ..-2 ..-3 ..-4 ..-5 ..-6 ..-7 ..-8 ..-g .

Since I A( G) is a torsion-free abelian group, this proves that

IA(G) is free abelian of rank 9.

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Now choose <p E InnG. By definition of InnG, there exists 9 x el ye2 Ze3 C~22 C~33 C~33 in G such that

<p(x) = g-1 xg , <p(y) = g-1 yg , <p(z) = g-1 zg .

By the normal form (see Eq. 7) we find that

and further use of Eq. 7 ultimately gives

<p(x) = xc~~cg, <p(y) = YC12elc~~, <p(z) = zc13elc23e2.

We obtain three specific elements of InnG by setting, in turn, ell,

e2 = e3 = 0; e2 = 1, e1 = e3 = 0; and e3 = 1, el = e2 = O. These are:

<P2(X) = XC12, <P2(y) = y, <P2(Z) = zc2i;

It is straightforward that

and that this expression is unique. This proves that

I nnG is free abelian of rank 3

and hence

InnG -# IA(G)

for G free nilpotent of class 2 and rank 3.

4.2. Remeslennikov's groups

In the category of nilpotent groups of class 4, consider Remeslennikov's groups:

and

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It is clear that Fs and Fr are also metabelian and torsion-free. Using symmetric bilinear forms (see §3.1 in Ref. 6), it is possible to show that Fs and Fr are not isomorphic but lie in the same localization genus. Corollary 2.1 gives that IA(Fs) and IA(Fr) are torsion-free nilpotent of class 3, and hence metabelian. It now follows from Theorem 2.2 that IA(Fs) and IA(Fr ) lie in the same localization genus.

4.3. Pickel-Roitberg groups

Pickel and Roitberg (see Ref. 4) associate a finitely generated nilpotent group of class 4 to a symmetric bilinear form as follows: let a : zn X zn -t Z be a symmetric bilinear form satisfying 0'( q, q) E 2Z for all q E zn. Let {e1' ... ,en} be any ordered basis of zn, and put

• O'(ei' ei) = 2bii ,

• O'(ei' ej) = O'(ej' ei) = bij for i =F j.

The n x n symmetric integral matrix B associated to this form has entries 2bii in the diagonal and bij = bji , when i =F j.

Two such forms are said to be in the same localization genus if they are equivalent up to a unit over the ring Z(p), for every prime p.

Let F = (x, y, Z1, ... , zn) be the free metabelian and nilpotent group of class 4, on n + 2 generators. Consider the group

where

G(O') = (x, y, Z1, ... , Zn; p) ,

p = IT [y, x, Zi, Zj]bij •

i~j

It is not hard to show that G is torsion-free if and only if the bij are relatively prime. Pickel and Roitberg proved that if 0'1 and 0'2 are in the same localization genus, then G1 = G(O'1) and G2 = G(O'2) are in the same localization genus. Assuming that the forms 0'1 and 0'2 give rise to torsion-free groups, it follows that G1 and G2 are torsion free, nilpotent, and met abelian of class 4. Theorem 1.1 now gives that IA(G 1 ) and IA(G2 )

are in the same localization genus.

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References

1. P. Hilton, Localization and cohomology of nilpotent groups, Math. Z. 132 (1973), 263-286.

2. P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, Notas de Matematica, North-Holland Mathematics Studies 15 (1975).

3. R. B. Warfield, Nilpotent groups, Lecture Notes in Mathematics 513, Springer­Verlag, (1976).

4. P. F. Pickel and J. Roitberg, Automorphism groups of nilpotent groups and spaces, J. Pure Appl. Algebra 150 (2000), no. 3, 307-319.

5. P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Math­ematics Notes, (1969).

6. M. Zyman, I A-automorphisms and localization of nilpotent groups, Ph.D. Thesis-City University of New York, (2007).

7. S. Bachmuth, G. Baumslag, J. Dyer, and H. Y. Mochizuki, Automorphism groups of two generator metabelian groups, J. London Math. Soc. (2) 36 (1987), no. 3, 393-406.

8. K. Maruyama, Localization of self-homotopy equivalences inducing the identity on homology, Math. Proc. Cambridge Philos. Soc. 108 (1990), nO. 2, 291-297.

9. E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equiva­lences, Topology 18 (1979), no. 3,187-197.