Cent roids

b~

Serguei Lioutikov

Department of Mathematics and Statistics SIcGill Univenit- Montréal

March 1999

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree

of Doctor of Pliilosophy

@ Serguei Lioutikov, 1999

National Library Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Selvices services bibliographiques

305 WeHiiigîon Street 395, nre Wellington ûtîawaON K 1 A W CXhwaON K 1 A W Canada Canada

The author has granted a non- exclusive Licence aîlowing the National Library of Canada to reproduce, loan, distribute or seU copies of this thesis in microforni, paper or electronic formats.

The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the author' s permission.

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de rnicrofiche/nlm, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Cent roids

b y

Serguei Lioutikov

Abstract

This thesis contains a description of centroids. or maximal rings of scalars.

for the claçs of CSA-groups, including free: universnlly free. and torsion-free

hyperbolic groups. CVe also describe centroids of free products of groups.

R-free nilpotent groups, and UT,(R), where R is a binomial ring. Csing the

results about the structure of centroids for R-free nilpotent groups, we solve

the long-standing problem of rigidity for R-free nilpotent groups. posed by

F. Grunewald and D. Segal. We also establish the rigidity of the groups

ml (R)

Cent roïdes

Serguei Lioutikov

Résumé

Cet te thèse contient une description des centroïdes, ou anneaiis maxi-

maux de scalaires, pour la classe des groupes CSA. y compris les groupes

libres, universellement libres, e t les groupes hyperboliques sans torsion. On

décrit également les centroïdes des produits libres de groupes, des groupes

nilpotents R-libres, et UT'(R), où R est un anneau binômial. En utilisant

les résultats à propos de la structure des centroïdes pour les groupes nilpo-

tents R-libres, on résout le problème de longue date au sujet de la rigidité

des groupes nilpotents R-libres, posé par F. Grunewdd et D. Segal. De plus,

on établit la rigidité des groups LrT,(R).

Acknowledgements

I would like to take this opportunity to express my deepest gratitude to

Professor Olga Kharlampovich, my thesis supervisor. for her helpful advice.

guidance. and encouragement.

1 greatly appreciate the invaluable help 1 got from Professor A.!dyasnikov.

who made t his thesis possible by his constant support. suggestions and ad-

vice. 1 would like to thank him for having introduced me to the interesting

problems treated in this thesis, and for his enthusiasm to discuss them and

to share his valuable ideas.

I am grateful to Fonds pour la Formation de Chercheurs et l'Aide $ la

Recherche and the CIChIA groiip for financial help.

Special thanks to my wife Ekaterina for her support throughout the prepa-

ration of this thesis. Her love and support have been a great source of moti-

vation for me.

Contents

1 Introduction 2

2 Definition of the centroid 10

3 Centroid as the maximal ring of scalars 16

4 Description of centroids of CSA-groups and free products of

groups 19

4.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . 19

4 . Centroids of CSA-groups . . . . . . . . . . . . . . . . . . . . . 2 1

4.3 Centroids in the category of -4-groups and esamples of simple

-4-groups with a given centroid A . . . . . . . . . . . . . . . . 27

5 Centroids of nilpotent groups 34

5.1 Centroids of free nilpotent groups . . . . . . . . . . . . . . . . 34

5.2 Centroid of L:T,(R) . . . . . . . . . . . . . . . . . . . . . . . . 47

6 The rigidity problem 51

7 Conclusion 53

1 Introduction

In the first part of this thesis for an arbitrary group G we introduce an as-

sociative ring with identity T(G), called the centroid of the group G. This

notiori is a natural generalization of the ring of endomorphisms to the case

of a non-commutative group G, and what is more important. it is an analog

of the notion of a centroid from ring theory. Shen we describe centroids of

various groups, in particular, centroids of free groups, torsion-free hyperbolic

groups, free nilpotent groups, and groups of unitriangular matrices LrT,,(R)

over an arbitrary associative ring R of characteristic zero. Finally. with the

description of centroids of free nilpotent groups in hand. we solve affirmatively

the rigidity problem for free nilpotent groups posed by F. Grunewald and D.

Segal in 1984 1121.

The most natural way to look at the centroid cornes from the theory of

exponential groups 11, 13: 18. 23, 2-11! i.e., groups admitting exponents in a

ring. Historically, the aviomatic notion of an esponential group grew out of

an attempt to solve the famous problern of A. Tarski: is the elementary theory

of free groups decidable? In other words, is there an algorithm determining

whether a given first-order closed formula is true in a free group'? Just re-

cently A. Myasnikov and 0. Kharlampovich announced a positive solution to

the problem of A. Tarski (see [16].) During the investigation of tliis problem

a particular case about the decidability of the positive %fragment of the ele-

mentary theory of free groups, and about the construction of sets defined by

positive 3-formulas, turned out to be of the greatest interest. These questions

are equivalent to the questions about the existence of an algorit hm determin-

ing the solvability of a systern of equations in a free group, and about the

description of the set of al1 solutions (or, in other words, the general solution)

cf such systerns.

In [18] R. Lyndon considered equations in one variable and proved the

algorithmic decidability of the question about the existence of a solution of an

equation of this kind. As a tool in his study, Lyndon introduced the notion

of groups with parametric exponents. which are now knomn as exponential

groups: and proved that the general solution of equations in one variable is

described by a finite set of parametric words.

Since then, systems of equations over a group have been widely studied (see.

for instance, [2], (111, [29]. [20], [30]) and is currently one of the main streams

of combinatorial group theory In 1980s G. bfakanin [?O] and A. Razborov

[29] proved one of the most signifiant results in this area: the algorithmic

solvability of systems of equations, and the universal theory, of a free group.

I t follows that systems of equations are also algorithmicdly solvable in 3-free

groups. (A group G is U r e e if, in the signature without constants, the class of

3-formulas true in G is the same as the class of 3-formulas true in a nonabelian

free group. Some authors prefer to talk about V-free groups, defined similady;

clearly, a group is 3free if and only if is V-free.)

In spite of the fact that Lyndon's methods gave only partial results towards

the solution of Tarski's problem, the concept of an exponential group turned

out to be of great independent interest. This interest was also rnotivated by

results of A. blalTcev[21], P. Hall[13], and G. Baumslag[l], but it was Lyndon

ho first introduced the a~ iomat ic notion of an exponential group in 1960.

In [23] and [24] .A. Blyasnikov and V. Remeslennikov developed a new refined

notion of an exponential group, adding one more wiom to Lyndon's definition.

This improved notion is more convenient because it coincides eractly with the

notion of a module over a ring in the abelian case. wliereas abelian exponential

groups after Lyndon constitute a far wider class. These two definitions coincide

in the case of free groups.

In the light of this nem airiomatic notion of an exponential group. wc prove

that i?(G) is the maximal ring of scalars for the group G: it means that for any

g E G, 7 E T(G) the esponent g7 E G is well defined, this action of T(G) on

G is faithful and satisfies some specific u ioms (including Lyndon's a'riorns).

and, moreover, r (G) is the maximal ring with respect to tliese properties. It

resembles the definition of the centroid T(K) of a ring K as the largest ring

of scalars of the ring K.

Constructively, the ring T(G) is defined as a subring of the near-ring P(G)

of al1 mappings of the group G. The whole study of the near-ring P(G)

began with the work of N. Fitting [3], where he investigated the set of al1

normal endomorphisms of G. H. Neumann [26, 271 used the near-ring P(G)

and its subrings in her investigations of varieties of groups. Later, A. Frohlich

[5, 6, 7, 81 applied near-rings of endomorphisms and their links with groups to

his study of non-abelian homological algebras. The results mentioned above

9 3 ~ fisc to numcrous attcmpts to construct somcthing similnr to "the ring of

endomorphisms" for an arbitrary group.

It is a well-known fact that over an abelian group G the set of al1 endo-

morphisms End(G) forms a ring with respect to the operations of addition +

and multiplication :

where g E G, 4, I,!J E End(G). For a non-abelian group G the set End(G) is

not a ring, because, for example. the sum of two endomorphisms of G is not

necessarily an endomorphism of G. Therefore, it is natural to consider more

genersl morphisms over G. It turns out that the set P(G) of al1 arbitrary

mappings from G into G forms a near-ring with respect to the operations (1).

The subnear-ring E(G) of P(G) generated by the set End(G) may seem to

be a good analog of the ring of endomorphisms of a nonabelian group G (in

the abelian case E(G) = End(G)), except that, in general, it is not a ring.

Many authors (1-1, 31, 25. 221 considered subnear-rings of E(G) generated by

different subsets (most often subsemigroups) of End(G). ÇVe define r(G) as the

5

ring which consists of al1 quasi-endomorphisms of G (i.e. such rnappings # thilt

(xy)" x4yyQ for commuting x, y E G) that centralize al1 inner automorphisms

of G (very sirnilar to the centroid in the ring theory).

In the sections that follow we give a detailed description of centroids of ar-

bitrary CSA-groups, including free groups and torsion-free hyperbolic groups.

Tlie clajs ijl CSA-groups was introduced by A. Xlyasnikoiov and 1.. Renieslen-

nikov in [23]. It plays a fundamental role in the investigation of -4-free groups.

tensor completions and groups with length functions. This class is quite wide:

for example: it contains al1 torsion-free hyperbolic groups, al1 groups acting

freely on .\-trees, and al1 3-free groups. In other words. the CS.-\-class com-

prises groups that are in some sense "close" to free groups. hlany results that

are valid for hyperbolic groups, or groups acting freely on :\-trees. can be gen-

eralized to this class. An important feature of the class of CS.-\-groups thst

makes it an object of g e a t interest in the study of centroids is the structure of

centralizers of CSX-groups. The key step in the description of the centroid of

a g o u p G is to describe the centralizers of nontrivial elements of G. The cen-

tralizers in CSA-groups possess certain properties that allow us to give such

a description. Namely, they are al1 abelian; the intersection of two distinct

centralizers is always trivial; and the centralizer of every nontrivial element is

conjugate separated.

A similar technique allows us to characterize centroids of free products of

groups. Surprisingly enough , the centroid T ( F ) of a free non-abelian group F

is quite big: r(F) is an unrestricted direct product of countably many copies

of 2.

In Section 4.3 we construct particular exponential groups with the centroids

prescribed in advance. To this end, let .4 be an arbitrary unitary associative

ring of charactcristic zcro and C? an A-group (sce Section 3 for definitionç).

Then one can define the notion of the centroid of G with respect to the giveri

action of -4 on G, Le., instead of r(G) one can consider the subring r .47

consisting of al1 normal quasi-endomorphisms from T(G) that commute wi th

the action of A. We prove that an arbitrary torsion-free hyperbolic group G

can be ernbedded into an -4-group G' such that Ta4(Ga) = .4 and al1 proper

centralizers of G' are pair-wise conjugate. In particular. if -4 is a field of char-

acteristic zero. then G* is a simple .A-group (no proper normal .A- subgroups

in Ge).

In Section 5 we describe centroids of finitely generated free nilpotent groups

and groups of unitriangular matrices UT&). In facto me prove the follow-

ing result: let !V be a non-abelian free nilpotent group of finite rank, R an

arbitrary binomial domain, and !VR the free nilpotent R-group introduced by

P. Hall [13], then the centroid I'(NR) is isomorphic to R $ 1, where I is a

nilpotent ideal of I'(G). The sarne also is true for the unitriangular ma t rk

group UT,(R) over an arbitrary associative ring R of characteristic O. CVe

then apply these results to solve the above-mentioned rigidity problem of free

nilpotent groups. Recall that a torsion-free nilpotent group G is rigid if for

any binomial domains R and S the Hall R-completion GR of G is isomorphic

to the Hdl S-completion GS if and only if the rings R and S are isomor-

phic. CVe prove that a non-abelian free nilpotent group of finite rank is rigid.

tvhich nnsivcrs thc F. Grunerald and D. Segal question from (1'7j. Noticc that

the same argument shows that the group UTn(Z) is rigid (which was already

proved by different methods in [Hl ) . Rigidity results form an essentisl part of

the F. Grunewald and D. Segal project airned to describe torsion-free nilpo-

tent groups up to isornorphism. Starting with a finitely generated torsion-free

nilpotent group G and a binomial ring R we can construct a new group GR

called the R-cornpletion of G (in the sense of P. Hall see [l3] .) In section 5 n e

give a description of the R-completion of a finitely generated torsion-free nilpo-

tent group G. Mal'cev proved that every torsion-free nilpotent group has a

Slal'cev ba i s (q, . . . , xh): a generating set such that each element is uniquel!

presented as xyl . . . x i h , where ui, . . . , ah are integers. We can associate to

every element in the group an h-tuple ( a l , . . . , a h ) . The group operations in

G are given by polynomials with rational coefficients. Now, if we choose ele-

ments a l , . . . , a h to be elements of some binomial ring R ( a commutative ring

R such that r(r - 1). . . (r - n+ l)/n! E R for each r E R and n > O ) and use

the same polynornials for multiplication Ive will end up with the group GR.

The R-completion of a finitely generated torsion-free nilpotent group is again

a torsion-free nilpotent group, though not necessarily finitely generated as a

Z-group. Provided that certain natural conditions are satisfied, GR will be

also finitely generated (see [12].) These observations led F. Grunewald and D.

Segal to propose the following classification project: for a given finitely gener-

atad tursiuii-frw idpoterit group G br different binomial rings R, tu classi-

the groups GR up to isomorphism as abstract groups. The definition of rigid-

ity arises naturally from the discussion above. It seems that centroids give an

appropriate tool to cleal with rigidity of nilpotent groups. Notice that in an

attempt to carry out this program one may consider centroids relative to the

category of nilpotent groups (see [25] for details).

The notion and properties of centroid. given in section 2. tvere first sug-

gested and studied by A. Myasnikov (unpublished.) Witli the exception of

several facts from section 4 concerning properties of CSA-groups, ivhere we

follow 1241 and [IO] and clearly acknowledge the sources. and the fact about

the structure of the centralizers of free nilpotent R-groups, which we suspect

to be known but give Our own proof due to the Iack of references, the ma-

terial presented in this thesis is new and constitutes original scholarship in

mat hematics.

2 Definition of the centroid

To the end of the section, let us tiu an arbitra. group G.

Definition 2.1 A set P with two binary operations + (addition) and . (mul-

tiplication) zs called a near-ring i f P is a group (no t necessarily commutative)

ri& r~spnct to the operation +. and n~ultzvlication is associati~ue and sutisfies

the left distributivity relation:

The lollowing is the principal example of a near-ring.

Let P(G) be the set of al1 arbitrary mappings of G into G. The sec P(G)

with the operations + and + defined by

f + V = 9"9", (y ut, t ) for g E G, <Plw E P(G).

is the near-ring of mappintjs of the group G.

Note that the mapping O : G -+ lc is a "zero'' of the near-ring P(G).

the mapping l p : g --t g is a "unit" of P(G), and the mapping -4 is defined

according to the rule g-@ = (g@))-l.

Definition 2.2 A mapping # E P(G) is called normal if

for al1 g , h E G. By N ( G ) tue denote the set of all normal mappings of G.

10

Clearly, iV(G) is the centralizer in P(G) of the set InnG of al1 inner au-

tomorphisms of G. In general, if M and X are subsets of P(G), then the

set

is termed the centralizer of X in M .

Let us denote by End(G) the nionoid of al1 endornorphisms of G with

respect to composition. Later on, subnear-rings will appear as the centralizers

of different subsemigroups of End(G).

Lemma 2.1 Let S 5 End(G) be an arbitranj semigroup of endornorphisms

of G. Then the centralizer CP(G) (S) of S in P(G) zs a subnear-ring in P(G).

Proof. Let f E S, & I/ E CP(G) (S) > g E G, then

Corollary 2.1 N(G) is a near-ring.

Lemma 2.2 Let 4, @ E N(G) , z, y E G. Then [z! y] = 1 implies [xm. go] = 1.

Proof. We have:

yx' = yx'y-lY = (yxy-L)#y = zoy.

Hence

Corollary 2.2 In the near-ring N(G) the operation of addition + is commu-

tative.

Proof. For g E G, 4, S, E N(G) we have:

Iience q5 + I ) = $J + 4. O

Lernma 2.3 For any set S E G the centralizer Cc(S) 5 G is .V(G)-

in-variant, i. e., V d E N(G)

Proof. According to lemma 2.2' for any x E .YT y E G, m E .V(G) the equality

[x, y] = i implies [z? z/@] = lT Le., c ~ ( S ) * ~ ( ~ ) C CG(~).

New-r ings with commatative addition are called abelian.

Question 2.1 Is it possible to realzze any abelian near-ring with 1 as the neur-

ring N ( G ) for some group G?

The following example shows that one can represent any abelian near-ring

with identity as a subring of N ( G ) for a properly chosen abelian group G.

Example 2.1 Let R be an abelzan near-ring with identity and G = R+ its

additive group. Then R acts on G by right rnultiplications: g -t ga, a E

112

R, g E G. This operation giues a rnonomorphzsrn R + P(G). Due to the fact

that the group G zs abelzan, any mappzng of G is n o n a l , ie . , P(G) = iV(G).

Corollary 2.3 There ezists an abelian group G for which X ( G ) is no t a n'ng.

Proof. Indced, according to the previous example, it is sufficient to take an

abclian ncar-ring with 1 which is not a ring. As an exaaple cf sucli near-

ring, we can choose the set of al1 integer polynomials Z[x ] with the standard

addition and composition as multiplication. 0

Definition 2.3 A mupping C$ E P(G) is called a quasi-endorrrorphisrn of the

gmup G if for any z, y E G we have:

[r. y] = 1 implies (zY)* = r'y?

The set of ull quasi-endomorphiams of G is denoted b y Q(G) .

Lemma 2.4 Q(G) U. a semiyroup with respect t o the composition of rnaps.

Proof. Indeed, for I, y E G, 9, # E Q(G) we have

[x, y] = 1 implies xmy4 = (xY)@ = = IJ%?

Hence

W. w [2, y] = i implies (zy)O'* = (zmyQ)* = x y ,

i.e., Q(G) is closed under composition.

Theorem 2.1 The set QG) of al1 normal quasi-endomorphisms of G is an

associatiue ring vzth 1 . I I zs celled the centroid of the group G.

Prao f. -4ccording to the definition, T(G) = N ( G ) n Q(G) is the intersection of

twvo semigroups, so T(G) is closed under multiplication. Let us verify that it is

closed under addition. Let (6,Q E r(G). We have already proved that X ( G ) is

a near-ring, hence 6 + w E N(G). Now it suffices to show that C$ + E Q(G):

i.e., 4 + 11, is a quasi-endomorphisrn of G. Let x, y E G, and [x, y] = 1. Tlien

But C#I and @ are normal, so by Lemma 2.1 the elements y@ and xW commute.

Therefore,

which, combined with the equality above, proves that 6 + u: E Q(G). Conse-

quently, r (G) is a near-ring; moreover, by Corollary 2.2 the addition in r (G)

is commutative. It remains to check that right-hand distributivity holds in

(again, here we used the fact that by Lemma 2.2 xm and x* commute). Since

this is true for an arbitrary x E C, we have (4 + $ ) O = #O + @O, and r (G) is

a subnng of P(G). Obviously, 1 E r(G). 17

There is an analogy between the construction of the centroid î ( K ) of a

ring E and the centroid T(G) of a group C. Namely, r ( K ) is the centralizer of

the set of a11 right-hand and left-hand multiplications of K in the semigroup

of a11 endomorphisms End(K+) of the additive group K+ of the ring K. Tak-

ing instead of the abelian group K+ a non-abelian group G, instead of the

semigroup End(G) its generalization Q(G), we obtain T(G) as the centralizer

of the set of al1 inner automorphisms of G in the semigroup Q(G). For the

present this anaiogy is only ari ariirlogy uf cuiistructiuii. i.e.. i t is purel? f~riiial.

But in the nest section ive will show that T(G) is the maimal ring of scalars

for G, i.e., r (G) satisfies the same universal property as the centroid of a ring

Observe that if the group G is abelian. then T(G) = End(G). i.e.. T(G) is

indeed a generalization of the ring of endornorphisms to the non-commutative

case.

Question 2.2 Whzck rings can be realized in the f o n T(G)?

3 Centroid as the maximal ring of scalars

Up to the end of this section, let us f i l an arbitrary associative ring with

identity -4. as well as a group G.

Given an action of -4 on G, i. e. a mapping G x .I + G, we will write the

result of the action of a E -4 on g E G as g". Consider the following auioms:

(gh)" = g"ha.

Definition 3.1 123, 241 The grovp G is called an -4-exponentiul yruup (or an

-4-group) if an action of the ring d on G sutzsfying aiiorns 1)-4) is defineil.

Notice that an arbitrary group is a 2-group; a group of period m is a. ZlrnZ-

group: a module over a ring .4 is an abelian -4-group and vice versa: D-groups

studied by G.Baumslag [l] are Q-groups; exponential nilpotent A-groups over

a binomial ring -4, introduced by P. Hall (131, are .4-groups; an arbitrary

pro-pgroup is a Zp-group over the ring of padic integers 2,.

The asiomatic approach to exponential groups first appeared in a paper of

R. Lyndon [18]. In his terms, A-groups are those that just satis& avioms 1)-

3). It turns out that this class of groups is too wide to work with; for example,

16

as we show belom, there are abelian A-groups in Lyndon's sense which are not

A-modules.

Example 3.1 Let B be a non-identity automorphism of the ring A, and .II a

free A-module wzth base (x, y). CVe can define o new action * of the ring A

on iCI as follows:

Then the action * of .4 on hl satisfies axiorns 1)-3), but if cro # 6) ( (uo) , then

hence aziorn 4 ) doesnP hold.

In fact, al1 the groups that Lyndon actually dealt with in his p a p a [18]

indeed satisfy aviom 4)' i.e., they are -4-groups.

Definition 3.2 Let G be an -4-group such that the action of .A on G is fuithJul.

Le., for any nonzero a E .S there exists g E G such that gu # 1. In this event

A is called a ring of scalurs of G.

The following proposition shows that G is a r(G)-group and T(G) is the

maximal ring of scalars of G.

Proposition 3.1 Let G be a group. Then:

1. the centroid r(G) i s a ring of scalars fo r G; in particular, G is a T(G)-

group;

2. T(G) zs the largest ring of scalars of the group G; i e . , if B is a ring of

scalars o j G, then there zs a canonical embedding B ct l?(G).

Prooj. For any t$ E r(G) and g E G we define g4 as the image of g under the

map 4. Notice that this action is faithful. tLvioms 1) and 2) hold by the very

definition of addition and multiplication in I'(G). k i o m 3) holds because ülI

mappings from T(G) are normal, and sviom 4) holds since r (G) consists of

quasi-endomorphisms of G. It follows that G is a l'(G)-group and T(G) is a

ring of scalars of G.

If B is a ring of scalars of G, then B acts faithfuily on G and Ive can define

a monomorphism of near-rings : B -+ P(G) tvhere 3 ( b ) : G -t G is the

mapping 9 ( b ) : g -t gb. By axioms 3) and 4) the rnapping @ ( b ) is a normal

quasi-endomorphism, hence @ ( B ) C T(G). 0

4 Description of centroids of CSA-groups and

free products of groups

In this section we describe the centroid of an arbitra- CS.-\-group, which

immediately gives us the structure of centroids of free groups, torsion-free hy-

perbolic groups, and groups acting freely on A-trees. Surprisingly, the centroitl

of a free non-abelian g o u p F is uncountable: T ( F ) = n,,,~, the unrestricted

direct product of countably many copies of 2. We then describe the structure

of the centroid of a free product of two groups G and H. I t turns out that

where the set of indices I is described in Theorern 4.2 below.

4.1 Preliminary results

In Our study of centroids we will frequently use the folloming important obser-

vation:

Lemma 4.1 Let G be an A-gmup. Then for any subset S of G the centmlizer

Cc(S) zs an A-subgmup of G, in particular, it is A-znuanant. 2.e.. for any

a E -4

Proof.

For any z E .Y, y E G, a E -4 we have

So, if [x,y] = 1 then [z,ya] = 1. 0

Wow, Ier us estahlish iin emh~dding of r ( G ) of an iirbitrary groiip C: in to

a certain Cartesian product.

Proposition 4.1 Let G be a n arbitrary group and let Ci (1 E 1) be fixed

representotives of the conjugacy clusses of the centralizers oJ al1 non-trivial

elements in G. Then there en'sts an embedding

Proof. Let Ci ( 2 E I ) be representatives of the conjugacy classes of the central-

izen of al1 non-trivial elements in G. By Lemma 4.1, al1 Ci are r(G)-in~ariant.

hence for every 4 E T(G) the restriction bi of Q ont0 Ci belongs to P(C,). Eacti

Qi is a normal quasi-endomorphisrn of Ci, therefore #i E T(Ci) . !doreorer. the

restriction map Ai : 4 -t di is a homomorphism of rings X i : T(G) -t r(C,).

This gives rise to the diagonal homomorphism

- where X = l l i E I X i . ÇVe daim that the homomorphism A is monic. Let O #

4 E I'(G). Then there esists an element g E G such that g" 1. Non?, some

conjugate heLgh of g belongs to Ci for some i and

g0 = (h(h-lgh))h-')" h(h-Lgh)Qh-L = h(h-Lgh)@ih-L # 1.

therefore, (h-Lgh)4i # 1 and, consequently, 4i # O. t?

If al1 the centralizers Ci in the proposition above are abelian. then I'(Ci) =

End(C,) and \Ive have the followinp

Corollary 4.1 Let G be a group in which al1 the centralzrers of al1 non-trivial

elements of G are abefian. Then there exists a n ernbedding

cuhere C, (i E 1) are representatives of the conjugacy classes of the centrutirers

of al1 non-trivial elements in G.

4.2 Centroids of CSA-groups

Definition 4.1 [ . 4 ] A group G is terrned a CS.4-yroup if al1 rnmirnal abelian

s~ubgroups in G are rn~lnorm~l.

Recall [17] that a subgroup H in a group G is malnormal if for any g E G

the condition Hg n H # 1 implies g E H .

Below ive collect some known results about CS.\ groups. To do this we

need the following

Definition 4.2 Subgroups -4 and B of a group G are called conjugate sepa-

rated if 4 9 n B = 1 for any g E G .

Proposition 4.2 [24] Let G be a CS.4-group. Then the following statements

are tme;

Any t uo rnuxirnul abelian subyroups of G either coincide or haue tnuial

intersection;

Any two murimal abelian subgroups of G are either conjupte or conju-

gute separated;

Commutation is an equivalence relation on the set o j al1 non-trioiul ele-

rnents of G;

The centralizer of any non-trivial element of G is a muxirrrul abelian

subgroup of G; conversely, uny maximal abelian subgroup of G # 1 is a

centralizer of any of its non-trivzal elements.

Now, we are ready to apply Coroilary 4.1 to CSA-groups.

Theorem 4.1 Let G be an arbitrary CSA-group. Then

where Ci (i E 1) are representatives of the conjugacy classes of the centralizers

of al1 non-trivial elements in G.

Proof. Let X : T(G) t ~ n d ( C , ) be the embedding from Corollary 4.1.

- CVe claim that X is onto. Choose an arbitrary 4 = lIiérk E Ti icr~nd(Cj) . Each

such 4 acts on C* as the endomorphism d i , and we can extend this action to

the union

of al! conjugatcs of C, by thc rulc

This action is well-defined, since the centralizers of non-trivial elements in G

either coincide or have trivial intersection. Notice that for any i # j we have

Zi n Z, = 1, therefore the union

gives rise to a well-defined mapping on G. From the definition of c one can

see that $ is a normal quasi-endomorphism of G. i.e. u E (G) . Obviously.

Corollary 4.2 1. Let F, be a free group on n generators. Then

2. Let G be a torsion-free hyperbolic group. Then

r(G) = ÏT& ;

23

3. Let F" be a free A-gmup (see [23, 241) over an associatiue un i tay ring

A of characteristic 0. Then r ( F A ) is an unrestricted Cartesian product

of infinitely many copies of the ring of endomorphzsms of t h e additive

group A+ of the ring -1:

Proof. In the case of a free group F it suffices to notice that a11 centralizers

C, of non-trivial elements in F are infinite cyclic. hence End(C,) - 2. If G is a torsion-free hyperbolic group. then the centralizers of al1 non-

trivial elements are infinite cyclic [9]. This implies. in particular. that G is a

CSh-group [24]. Now the statement follows from the theorem above.

In the paper (241 the free A-group Fq4 over a ring .4 was described in terms

of HYN-extensions. I t was proved there that F;' is a CSA-group, and that the

centralizers of non-trivial elements are isomorphic to the additive group Ai of

the ring -4. 0

The following theorem clarifies the structure of the centroids of free prod-

ucts of groups.

Theorem 4.2 Let G and H be arbitrary groups, then

vhere the unrestricted direct product zs taken over al1 the conjuyacy classes of

the centralizers of elements in G * H which are not conjvgate to on element of

PTOO~. If G or H is trivial then the statement of the theorem is obvious. So

we can assume that both G and H are non-trivial.

We begin by recalling several properties of Free products of groups (see

details in [17]).

It follows from the Kurosh Subgroup Theorem that every proper centralizer

in G* H is either infinite cyclic or conjugate to a centralizer in one of the factors.

From The Conjugacy Theorem for Free Products (see (171) we have that

G and H are malnormal and conjugate separated in G * H.

Let {C'Ji E I ) contain one representative of every conjugacy class of the

centralizers of elernents in G * H which are not conjugate to an element of G

or H. Kotice that if i # j then C, nC, = 1. Indeed. suppose C, = g p ( z ) . C, =

gp(y) and 1 # z E C,n C,. then the subgroup g p ( x . y) has a non-trivial center.

and hence by the subgroup theorem g p ( x , y) is cyclic. That implies that x and

g cornmute, hence C, = C,.

Define a homomorphism of rings

as follows. Let # E T(G) x r ( H ) x nier2. Then 4 = (O,& f ) where o E

r(G), 6 E r ( H ) , f E TÏiEIZ ( we think of f as a function f : I -+ 2). Now we

define a rnapping c(4) : G t H -t G * H according to the following three cases.

Take an arbitrary g E G * H. Then either g is in a conjugate of Ci for some i or

35

it is conjugate to some element in G or else it is conjugate to some element in

H. Suppose first that g is in a conjugate of some Cio. As Ive mentioned above.

this g does not belong to any other Cj, therefore the number io is uniquely

determined by g. In this case put

IF y is in a conjugate of G, Say g = z- laz , ( a E G), then put

Similarly, if g is in a conjugate of H, say g = z- lb i , (6 E H), then define

The mapping f(d) is well-defined and by construction it is a riormal quasi-

endomorphism, hence <@) € T(G * H). One can check directly that < is a ring

homomorphism.

We claim that ( is an isomorphism. To show that is onto' consider an

arbitrary mapping I) E T(G * H ) . For any non-trivial g E G the centralizer

C(g ) of g in G* H is contained in G; since this centralizer is r(G * H)-invariant.

ive have that g* E G, and consequentl- GY> c G. Thus the restriction *WC; of y,

to G belongs to r(G). Similarly, the restriction QH of P to H belongs to l?(H).

Again, each centralizer Ci is $-invariant and cyclic; therefore the restriction

of 6 to Ci acts on Ci as the multiplication by a given integer, say ni. Define

a function fy, : I + Z by f&) = ni. NO^, for @ = (QG, qH, /#) we have

c(Q) = $. Thus { is onto.

To show that ( is a monomorphism consider an arbitrary

and assume that f(#) = O. Then a = 0 , d = O, f = O. i.e., ç5 = 0. 0

4.3 Centroids in the category of A-groups and examples

of simple A-groups with a given centroid -4

Let G be an A-group over an associative unitary ring -4. We begin this section

by defining the -4-centroid rA(G) of an .4-group G, i.e.. the centroid with

respect to the category of .4-groups.

Definition 4.3 Let G be an -4-group and CD : .A -+ T(G) the canonicul em-

bedding of -4 into the maximal ring of scaiars T(G) (see 3 1). Denute bg

Tm4(C) the centrakzer in r (G) of the set @(A).

Xotice that r 4 G ) is a subring of T(G). If the ring -4 is commutative. then

r A ( G ) is an algebra over A.

If G is a CSA A-group then every proper centralizer C of G is an abelian

A-group, i.e., a module over the ring A. In this event the ring End&) of al1

A-endomorphisms of C is well-defined. Now we are in a position to formulate

the following result .

27

Proposition 4.3 Let G be a CSA -4-group. Then

vhere Ci (i E 1) are representatives of the conjugacy classes of the centralkers

oJ all non-trivial elenzents zn G.

ProooJ. As we have meritioned already, the ceritraiizers Cl are abelian -4- groups.

i.e., they are .A-modules. Hence every mapping # E rA(G) induces an -4-

endomorphism on Ci. Yow the result follows from Theorem 4.1. CI

Let G be a torsion-free CS-\ group and .-L an associative unitary ring of

characteristic zero (i.e., the additive group AC is torsion-free). Then the group

G can be embedded into an A-group G" which is called the -4-completion of G

(see (241). The group G" can be defined by a universal property similar to the

tensor extensions in the category of modules. namely, there exists an embed-

ding X : G -t Ge4 such that for any -4-group H and any -4-homomorphism

Q : G + H there exists an -4-homomorphism 4"' : G;' -t H such that

@a4 O X = 9. Now we are able to formulate another corollary of Theorem 4.1.

Corollary 4.3 Let G be a torsion-lree hypetbolic group, -4 be an associative

unitary ring of characteristic zero and G-4 be the -4-cornpletion of G. Then

an unrestricted Cartesian product of copies of the ring -4.

Proof. Let G and .4 be as above. In this case the completion GA is a torsion-

free CSA group in which the centralizers Ci are free cyclic A-modules [%].

Therefore, Enda4(Ci) 2 A, and the result follows from Theorem 4.1. CI

Let F be a non-abelian free group and A as above. The next result shows

how to embed F into a non-abelian CSA .&group F' such that Te4(FB) = -4.

ail proper ceritraiizers in F' are isoriiorphic to ri', arid ail of theni are painvise

conjugate. To construct F' we do the following. The first step is to extend

al1 cyclic centralizers of F to be isomorphic to the group =Li. The second

step is to pair-wise conjugate al1 the centralizers isomorphic to .-Li by means

of HXN-extensions. In this step new cyclic centralizers occur, so me repent

consequently steps 1 and 2 countably niany times. The resulting group will

satisfy the conditions mentioned above. To car- out this process we need.

first. to justify that one can extend the centralizers up to .A+ without losing the

property of being a CS.\-group. This was done in [24] and the corresponding

construction is called the A-completion of the group. Secondly, we have to

show that one can indeed conjugate the centralizers staying inside the class

of CSA groups. This was proved in (101. To explain ive need the following

definition.

Definition 4.4 Let G be a group and 4 : -4 -+ B an isornorphzsm of subgroups

of G. The HNN-extension

is culled sepurated if the subgrovps A, B are conjugate separated in G (see

Section 4.2).

Now we can formulate the following result.

Theorem 4.3 [IO] Let G be a torsion-free CSA group and

a separated HNN -extension of G with abelian associated subyroups -4: B. If

at least one of the subgroups -4, B is maximal abelian. then H is a torsion-free

CSA-group.

Also we need the following result.

Proposition 4.4 [24] .4 direct limit of torsion-free CSA groups is a torsion-

free CSA -group.

Xow we are in a position to prove the following theorem.

Theorem 4.4 Let F be a free group und -4 be a n associative unitanj ring of

characteristic zero. Then F is embeddable into a CS.4 .A-group F* for which

the following conditions hold:

1. al1 proper centralizers of F* are isomorphie to A+ and pair-wzse conf i -

gate;

Proof. To prove the theorem, it suffices to embed F into a CSA-group F'

such that al1 proper centralizers of F' are isomorphic to .Af and conjugate to

each other. Then by Theorem 4.1 and Corollary 4.3 Ive will obtain 2) and 3)

respec thel y.

We start with describing the following construction. Let G be a torsion-

free CSA group. By Proposition 4.2, any two proper centralizers . I I . .Y of G

are either conjugate to each other or conjugate separated. Denote by JU the

set of representatives of conjugacy classes of all proper centralizers in C; which

are isomorphic to P. Let {JI& < 6) be a well-ordering of the set M. Put

Go = G and define groups G, by induction. If Ga (9 < 6) is alreatly defined

then Gj+l is equai to the following HNN-extension

For a limit ordinal X ( A < 6 ) define

Observe that for eacli /3 the centralizers Mo and are not conjugate

in the group Gp (see, for example, [ l ~ ] for a description of conjugate elements

in HNN-extensions), hence if the group Go is s CSA-group then Mo and Ma

are conjugate separated and both are still centraiizes of the corresponding

elements from Gg (in particular, they both are maximal abelian in GD). In

this event, by Proposition 4.2, the group GBci is again a torsion-free CS.&

group. Since a direct limit of torsion-free CS;\-groups is also a torsion-free

CSX-group, then by induction al1 the groups Ga (a < 6) are torsion-free

CS.-\-groups. Denote

The group is a torsion-free CS.1-goup in which al1 the centralizers from iM

are pair-wise conj ugate.

Now define by induction the following chain of groups. Put Fo = F and

suppose that for sornc positive integer i the group F, is already defined. Sup-

pose also that al1 proper centralizers in Fi are either isomorphic to .A+ or

infinite cyclic. Since A is a ring, it acts by multiplication on A+. We turn

F, into a partial -4-group (see [24]), letting A act by multiplication on al1 the

centralizers from Fi which are isornorphic to .A+. Let F t be the .-Lcornpletion

of the partial -4-group Fr (see [24]). Shen F;" is a torsion-free CS.4-group in

which al1 proper centralizers are isomorphic to A+. Moreover, any centralizer

in Fi which is isornorphic to .Af is still the centralizer of the same element in

the group F,". Let

By induction and the argument above, al1 the groups Fi are torsion-free

CSA-groups, as ive11 as the group

By the construction, al1 proper centralizers in F* are isomorphic to .4+ and

pair-wise conj ugate.

Corollary 4.4 If -4 is a field of characteristic zero? then the yroup F' from

the theorem above is a simple -4-group, i.e., there are no proper normal A-

subgroups in F* .

Indeed, suppose N is a proper normal -4-subgroup of F* and z E F* - -V.

y E N . Then the centralizers C(z) and C(g) are conjugate in F a . Since

proper centralizers in F' are cyclic -4-modules and .4 is a field, C'(y) 5 3.

Hence C ( r ) 5 'i and x E :V - a contradiction.

Remark 4.1 One can replace the free group F in the theorem uboue b y un

arbitrary torsion-free hi~perbolzc grovp G and the theorem still holds (the same

proof l

5 Centroids of nilpotent groups

5.1 Centroids of free nilpotent groups

Let G be a group. By

G = G l 3 G 2 1 . . .

we denore the lower centrai series oE G, here Gi+, = ici. Gj. The upper cencral

series of G is denoted by

where Zii1 (G) is the preimage in G of the center Z(G/Zi(G)) under the canon-

ical epimorphism G + G/Zi(G). The subgroup &(G) is the center of G and

we often write simply Z(G) instead of Zl(G).

We begin with the following lemma.

Lemma 5.1 Let G be an urbitrary group. For every g' h E G! $ E T(G) such

that [g , [g, hl] = 1 the followiny equality holds

Proof. Observe that if [g, [g, hl] = 1, then g and h-lgh cornmute (since h-'gh =

g[g, hl). Therefore,

Corollary 5.1 Let g, h E G, 4 E r(G). If [g, h] E Z(G) then

[gm! hl = [g, h'] = [g, hl".

Corollary 5.2 T h e followzng statements are true for an arbztraq group G:

Proof. If z E Z(G), then [ z , g] = 1 For any g E G. Hence by the lemma above

for any @ E r(G) we have [zQ, g] = [s,glQ = 1. tlierefore 9 E Z(G). This

proves 1).

By definition Z2(G) = ( 2 E Gl[r,g] E Z(G) for every g E G} . By the

lemma above for any 2 E Zz(G). <I E G. and 4 E r(G) we have

which shows that 2' E Z2(G). 0

From now on we will assume that G is a finitely generated non-abelzun free

nilpotent group of class c wzth baszs X I , . . . , r,. In this event Gi = Zcdi+l (G)

for each i = 1,. . . , c and each subgroup Gi is isolated in G, i.e., for any g E G

and any integer n # O, if gn E Gi, then g E Gis

Notice that every non-trivial element g E G has a unique maximal root in

G, i.e., the unique element go E G such that g = g?, where m is the greatest

positive integer for which the equation g = xm has a solution in G.

35

The following proposition is assumed to be known, but we need the proof

to be able to describe centralizers of elements in an arbitrary free nilpotent

R-group at the end of this section.

Proposition 5.1 Let G be a free nilpotent group of class c. If g E Gi \ Gi,i,

then

where go is the maxzmal mot of g rnodulo (und go = 1 if g E Gc-iTl).

Proof. Let g E G, - G i + i , u E G, -G,+,, and [ g . o ] = i. If j 3 c - i + 1 .

then u E (go) -Gc-l+l. Suppose now that j < c - i + 1. Then by corollary

5.17 in [19] the following holds: i = j and for some element w E G, n e

have gG,,l = wPGlil a11d L . G ~ + ~ = uqG+, . I t follows that 9%-P E GL;l

and still [g,gqudp] = 1. By the argument above g q ü p E Gc-L+l. and hencc

9'Ge-i+i = v P G ~ - ~ + ~ . The nilpotent group G/Gc-L+I is torsion-free. therefore

the canonical images of the g and v in G/Gc-t+l are powers of one and the

same elernent, so they are powers of goGc-i+i This implies that u is a power

of go rnodulo Gc-i+t, as desired.

cl

Leinma 5.2 For every p E T(G) there exists n, E Z such that

1. if g E G, but g $ G2, then gr = gnr rnodulo Z(G);

Proo f. We show first t hat x: = x n g for every basic element xi . By Proposition

5.1

Cc(zi) = ( x i ) Z(G) CG(zixj) = ( x i x j ) Z(G)

for every i, j = 1,. . . ,m. Fix an arbitrary E T(G). Since centralizers of

elements are ï(G)-invariant, for e v q i, j = 1,. . . , m we have:

for some integers ni, nil and some elements Zi, q j E Z(G). Let US show that

nt = ni for al1 i = 1,. . . . m. Indeed, let

c-2

Then u is a basic commutator of weight c - 1. Then by corollary 5.1

On the other band,

Since lu , x i ] and [IL, x l ] are basic cornmutators, we deduce t'rom 2 and 3 t hat

ni = ni1 = nl. Put n, = nl.

The next step is to prove that zY = Y"' for every 2 E Z(G). Since Z(G)

is an abelian group, it suffices to show that this equality holds for generators

37

of Z(G). The center Z(G) is generated by simple cornmutators of weight c in

the generators X I , . . . , xn. We have

which proves the statement.

Now we show that q î = cf9 modulo Z(G) for al1 elements q in G but

not in G2 which are not proper powers modulo Z(G). By Proposition j. 1.

Cc(g) = (9) fl Z(G). Thus, gv = g"y.9 - 9 , ~ : where n,,, E Z and z,,, E Z(G).

We need to show that n,,, = n,. Since g G2, there exists an element

u E &(G) such that 1 # [y, u ] E Z(G). In this event [g. u ] p = [g. u ] ' ~ . On the

other hand,

Since the center of a free nilpotent group is a free abelian group, this implies

that n , , = n,.

Now let g E G be an nrbitrary element which does not belong to G2. Then

g = g,kz, where go is not a proper power modulo Z(G) and r E Z ( G ) . Then.

which completes the proof of the lemma.

Definition 5.1 Let

I = { y E r(G) 1 n, = O}.

35

The center Z(G) is a r(G)-subgroup of O, hence for every 4 E T(G) the

restriction 4' of 4 to Z(G) belongs to I'(Z(G)). Denote by

the corresponding ring homomorphism ~ ( 4 ) = 4'. The definition above implies

thnt I = k e r ! ~ ) .

Notice that every integer n gives rise to a mapping un : g + yn which

belongs to T(G). We will identify the ring of integers Z with the corresponding

subring in r(G) under the embedding n + @,. Now we can formulate the

following

Lemma 5.3 I 2s an ideal in T(G) and T(G) E Z @ I .

Proof. We have mentioned already that I = k e r ( r ) . hence I is an ideal in

r (G) . So we need only to prove that T(G) = Z @ I .

By definition, p E I if and only if n, = O? so Z n I = O. Let E T(G) and

I E Z(G). Then

-V-% = y', . p - 3 = 1-

Therefore p - n4 E 1, and T(G) = Z $ I .

Lemrna 5.4 Let g be an arbitrary element from Gi svch that g 6 Gi+l. Then

the following hold:

Proof. We prove the first part of this lemma by induction on i. If i = c, then

the statement holds by lemma 5.2, part 2. Let i 2 cl2 and let g be an arbitrary

element from Gi which does not belong to G;+I. Then for every r E G ive

have [ g , x ] E Gicl and hence by induction

Observe that g-l commutes with X - ~ C J X since

Therefore

This implies that

Thus, for any x E G we have [g4-"',XI E Gii2< and consequently, g d - " ~ E

Gi+ 1. Part 1 of the lemma follows.

Now, let us prove the part 2. Suppose i < c/2 . For any r E Gc-i we have

[g, x] E W), so

This implies tha t gp-". E Gici Observe that [ g , g Q - n ~ ] = 1 , hence by

Proposition 5.1 we have

(we used here that i < c/2 and hence i + 1 < c - i + 1).

Corollary 5.3 Lct G be a free nilpotcnt group of finite r a d . The" Z,(G) k

T(G) -invariant jor euch i.

Theorem 5.1 If C is a non-abelian free nilpo tent group of class c, then T(G) =

Z @ 1, and 1% O, where d 2s the smallest integer such that d > ci'>.

Proof. This theorem follows directly from Lemma 5.3 and Lemma 5.4. 0

Analyzing the proof of the theorem above, one can see thac the argument

works also for free nilpotent R-groups in the sense of P. Hall [13]. Recall that

an integral (commutative) domain R of characteristic O is termed a binonriul

dornain if for any r E R and n > O the equation

hos a solution in R. Now, following P.Hall (131, we describe the R-completion

GR of an arbitrary torsion-free finitely generated nilpotent group G.

An ordered set of elements al,. . . , a, E G is called a hlal'cev basis for G if

every element g of G c m be uniquely exptessed in the form

where t l (g) , . . . , t ,(g) E Z and the subgroups Gi = < ai,. . . , a, > form a

central series

in the group G. The numbers ti(g)'s are called the coordinates of g with respect

to the given basis al . . . . a,.

A. XIal'cev [X] proved that such b a i s exists in every torsion-free nilpo-

tent group G (see also [13]). Moreover. he proved that the multiplication

and exponentiation in G can be defined coordinate-wise by some polynomials

f i ( l i : . . . , ~ , , y l , . . .,y,) and hi (z l , . . . ,ln? y,:. . . , y n ) (i = 1.. . . . n) with ra-

tional coefficients. 'iamely, for any elements u? v E G and an integer X the

following holds for each i = 1,. . . , n

t,(u.c) = f i( t (u), . . . . t,(u), t &). . . . . t,(,c)).

Xow let

where aiL . . .an is just a formal product of this type.

If u = ai1 . . . a n E GR, then elements t i (u) = ri E R ( i = 1? . . . , n) are

called coordinates of u. Now Ive can define multiplication and R-exponentiation

on GR by the formulas (4) and (5) (assuming in the latter that X is an arbitrary

element in R). This turns GR into a nilpotent R-group. Notice that if N is a

free nilpotent group of class c with basis 11,. . . , x,, then N R is a free nilpotent

R-group (in the P.Hall category of nilpotent R-groups) of claçs c and with basis

X L , . . . : x*.

Analogs of Proposition 5.1 and Lemmas 5.2, 5.3 and 5.4 also holcl for

the free nilpotent R-group G = N R . To explain this. denote by F the field

of fractions of the intcgrd domnin R. It follons from the P.Hall construction

above that G = N R canonically ernbeds into H = .VF. The same argument as

above implies that if g E Hl - Hl,I then

This translates into the group G = :VR as follows: if g E G, - G,,i and

[ g , u ] = 1: then either u E G,-,+l or g"G,-,+l = U ~ G , - , ; ~ for sorne a. 3 E R.

Observe that another way to prove this is to use SIal'cev's correspondence

betrveen free nilpotent F-groups and free nilpotent Lie F-algebras (see for

example [32]) as follows.

Proposition 5.2 Let I ï be the Jield of~ractioris of an integral domuin R. Let

L be a free nilpotent 1;-algebra of deyree c and E L,' then for an element

h E L tue have ( g , h) = O if and on$ if h = og modulo Lc- i+ l , for some

a! E K.

Proof. Fve mite g = g, + gi+l + . . + + g,, where gk is a homogeneous

component of degree k for al1 i <_ k 5 c and also h = h, + hj,l + . . . + h,.

43

Then (9, h) = ( g i , hi) + 6(g1 h) , where 6 ( g , h) denotes components of higher

degree. This implies that (g i , hj ) = O. It is a known fact (see [19]) that if d

and $ are homogeneous elements of a free Lie algebra over K , then ( r p , o) = O

if and only if 4 and S, are linearly dependent over K. Therefore, we have that

either i + j > c or i = j and hi = a g i for some a E K. Now we will investigate

thc lattcr c s c . WC considcr a ncw clcmcnt f = h - ag. Sotc tha t (g. f) = 0.

We may write f = fk + fk+l + d(/) , where d( f) denotes components of higher

degree and k > i (the ith term gets canceled.) Now. (g? f) = (gi, fk) + d(g. f).

where 6(9. f) denotes cornponents of higher degree and we know that k > i.

This implies that i + k > c, hence h, = ag, for al1 i 5 s 5 c - i. which

concludes the proof. 13

Proposition 5.3 Let K bc the field of fractions O/ an integml doniairl R. .Y

a free nilpotent group and H = Ji". If g E Hi \ HL+,. then

Proof. We twill use 'vlal'cev's correspondence between free nilpotent Lie K-

algebras and free nilpotent I\'-groups (see, for esample, (321.) Take a free

nilpotent Lie algebra L over K. Consider a new operation * on L, given by

the Baker-Campbell-Hausdorff formula (see (191):

where (g,h) represents the Lie multiplication in L and 6 stands for Lie products

of higher degree.

According to [X], the set L equipped with the operation * forms a Free

nilpotent II-group M = (L; *). Exponentiation in 1L.I is given by ga = ag.

where g E L and a E K .

From the Bal<cr-Campbell-Hausdorff formula it follons thnt tno clcmcnts

g, h E M commute if and only if (g, h) = O in L. Now the statement of the

proposition follows from the description of commuting elements in a free Lie

algebra. n

Xow, the argument in Lemmas 5.3 and 5.4 goes through without any

changes. And we can formulate the following facts.

Let us show how to prove the analog of Lernma 3.2. To formulate the

analog! one needs just to replace n, E Z by n, E R. Then the first part of

the proof (i.e.. n, = nl = n, for al1 i ) works readily for the analog. To prove

that 19 = zn7? observe that the argument in the lemma holds if we can prove

that each 9 E r(G) acts on Z(G) as an R-endomorphism. To show this. it

suffices to prove that p acts as R-homomorphism on R-generators of Z(G). for

example, on al1 simple commutaton of weight c in the generators LC 1 7 . . . . r,, .

Now for any r E R the following holds:

which proves the staternent and the second part of the lemma. To finish the

proof, let us consider an arbitrary g E G - G2. Since the centralizcr of g in G

is invariant under I'(C), we have that

for some a, ,d E R and i E Z(G). Since g $ G2, there exists ari element

u E Z2(G) such that 1 # [g, u"] E Z(G). In this event

[g , u a y = [g, XP]"' = [f', ua] = [$ q q a .

On the other hand,

Since the center of a free nilpotent R-group is a free abelian R-group (i.e.. free

R-module), this impiies that an, = j. But then

and since R-roots are unique in the quotient group G/Z(G) we have gr =

gng rnodvlo Z(G), as desired. This finishes the proof of the lernma.

Çummarizing the discussion above, me have the following result .

Theorem 5.2 Let R be a binomial dornain and GR a finitely generated non-

abelian free nilpotent R-group of class c. Then r (GR) = R @ I , and IL O ,

vlhere d is the srnallest integer such thut d > c/2.

5.2 Centroid of UT,(R)

Now we will prove that the centroid of G = UTn(Z) , n 2 3, hm a structure

similar to the structure of the centroid of a finitely generated free nilpotent

group.

By t,, i < j we denote the elements of G having 1's on the main diagonal

and in the colurnn j of the row i, and 0's everywhere else.

The center of G is the cyclic group generated by tL,. Since the center of

G is Linvariant, we c m define a ring homornorphism T : T(G) + 2.9 ct n,.

where n, f Z is such that tyn = t?.

Note that any integer n can be viewed as an element of T(G): n takes

elements to their nth power.

Lemma 5.5 T(G) zz Z $ I , i~drere I = k e r ( r ) a r (G) .

Proof. Any y T(G) can be written uniquely in the form y = nr + (; - n,J.

where n, E Z and q - nn, E 1. Mso. it is easy to see that Z n I = {O}. which

çompletes the proof. C!

Proof. The elements t , constitute a hlal'cev basis for G, and every element

g E G can be written uniquely in the form: g = t:i2ttF .. . taln L n . Let us 6s

an arbitrary element cp E I and m i t e the image of g under p in the form:

. , . t . We will prove the Iernrna by çhowing that if ai, = O for g' = t12 t23

al1 j : 1 < j < N and for al1 i : 1 5 i < j, then pij = O for al1 j : 1 < j 5 N

and for al1 i : 1 <_ i < j .

Before we begin the proof. let us point out the following useful identity:

First, let us assunie that al? # O. W e can think of g as g = ab. where

a = ta" ,, and b = t P 3 t : t 4 . .*tzn. Then. using identity 6.

Since all factors t;' in b are such that i < n and j > 2. we have [b; t.,J = 1. so

but tyc :? Z(G), hence [g, t2,] = t?: E Z(G) . Then by lemrna 5.1 1 =

Suppose now that Q i j = O for al1 j with 1 < j < N and al1 i rvith 1 5 i < j .

For j and i as above, we will prove by induction on j that d j = O. By the

induction hypothesis, fiik = O for al1 k. < j , i < k.

FVrite g+' in terrns of the Evlal'cev basis: g'J = . . By the

induction hypothesis, the first non-trivial term in this product is t;~;;; let us

denote it by al and the rest of the product by b l . Then by the second part of

identity 6, we have

- $ J - ~ * J rt4-1,, b ~ f b + 1 1 - ' , -~ ,n L ~ - 1 , n 7 i j i 17

Note that the second Factor vanishes due to the lact that bl does not contain

tZk with k c j by the induction hypothesis. Thus we simply have

DJ - 1.1 1 = tj-,,, [bl' t j n ] *

We now turn our attention to the commutator [bl: t,,]. Write bl = a2t,-2,,b2.

where a? cornniutes with tjn. üsing identity 6 again? we see that

by the same argument as before. This yields

Pj t , j t f j - 2 , , 1 = tj-<+n 1-2.n [ b ~ , tjn]

Proceeding in the same fashion, we obtain

Because the elements t i j constitute a Mal'cev basis for G, this implies that

Oij -i,= O For al1 i < j < iV.

- t P . ~ - i H P.Y-z N QI.v Note that [y, t lVn] - N-1 n tiV-* n tln . hIoreover, [ t l j , [go tAvn]] = t:~" E

Z(G). Since [gl [g, tNJ = 1, by lemma 5.1 we have 1 = [ t l j ? [9: t lvn]]*^ =

[tu, [$?, tNn]] = tp . Therefore, hiV = O for every j < 3. This concludes the

proof of the lemma. C!

Theorem 5.3 If C = UTn(Z), then T(G) = Z $ I , and In = 0.

Proo f. This theorem follows directly from lemmas 5.5 and 5.6. O

It is easy to see that lemmas 5.5 and 5.6 continue to be valid if vie replace

the ring of integers Z by an arbitrary associative ring R of characteristic O (the

çame argument). This allows us to formulate the following theorem:

Theorem 5.4 If G = L'Tn(R), then T(G) = R $1, and In = O.

6 The rigidity problem

Definition 6.1 A torsion-free nilpotent group G is called rigid if GR GS

implies R E S for any two binomial domains R and S of churacteristic 0.

In [12] F.Grunewald and DSegal proved that UT, (Z) ( n 2 3) is rigid. and

formulateci the following problem:

Problem 6.1 Is a finitely genemted non-abelian free nilpotent group N rigidY

Gsing the structure of the centroid of the free nilpotent R-group NR ab-

tained in the previous section, we can answer this question affirmatively.

Theorem 6.1 Euery finitely generated non-abelian free nilpotent group .V 1s

Proof. Suppose that XR z .VS. Then their centroids are also isomorphic.

ço T ( N R ) z r(XS) . and from our description of centroids we obtain that

R @ II E S $ I2 where Ii and 1 2 are as in theocem 5.4.

We now claim that I I is an ideal containing al1 nilpotent elements in r ( N R ) .

Indeed, al1 elements of II are nilpotent by theorem 5.4. If r E R, r # O. then

(r f i ) " = rn + j # O, since rn # O. Similarly, 1 2 contains al1 nilpotent elements

of r (hiS).

Denote by / the isornorphism between Re I I and S 8 12. Since the image

of a nilpotent element under f is again nilpotent, f (II) = 4. Therefore, f

induces an isomorphism f : R $ Ii/Ii + S $ I&, which implies R = S. O

51

Finally, ae observe that according to theorem 5.4: the centroid of the

group UT,(R) is isomorphic to R @ 1, where 1 is a nilpotent ideal. Moreover,

u T , ( Z ) ~ = UT&?). Thus Our rnethod offers an alternative proof of the fact

that U T J Z ) is rigid for every n 2 3.

7 Conclusion

The results about rigidity of finitely generated torsion-free free nilpotent groups

obtained in this thesis present an important step tomards the classification of

finitely generated torsion-free nilpotent groups. Until now only few results

about rigidity of such groups were established. In (121 F. Grunewald and D.

Segal proved that the groups UT,(Z) are rigid for al1 n 2 3. And the ques-

tion of rigidity of any other irreducible finitely generated torsion-free nilpotent

groups, including free nilpotent groups, was open. It appears that the notion

of centroid may turn out to be a very appropriate tool for solving this prob-

lem. We have a reason to believe that the problem of rigidity for 2-nilpotent

groups can be solved using a technique similar to that used in this thesis.

This will help to further develop the classification project for finitely gener-

ated torsion-free nilpotent groups started by F. Grunewald and D. Segal. The

question about the rigidity of a finitely generated torsion-free nilpotent group

G of higher degree is, in general, more difficult. It seems that in this case

it is reasonable to exploit the centroid related to the category of nilpotent

groups r w ( G ) , which is defined to be the maximal subring of T(G) satisfying

the following properties: (i) the terms of the lower central series should be

rN(G)-subgoups and (ii) for every elements g E Gi, h E Gj and 4 E r N ( G )

the following identity should hold

Centroids could be also useful in answering some mode1 theory questions

about nilpotent groups. There are two important hypotheses about elementary

equivalence of finitely generated nilpotent groups stated by V. Remeslennikov

and V. Romankov in 1983:

Hypothesis 7.1 Let G and H be elementarily equivalent finitelg generated

nilpotent groups and let Z(G) 5 G'. Shen G und H are isomorphic.

Hypothesis 7.2 E lernen tady equivalent Jinitely generated nilpotent groups

uiithout torsion are isomorphic.

These hypotheses lead to a variety of less general problems sorne of which

could be treated using centroids.

References

[l] Baumslag G. Some aspects of groups with unique roots. Acta Math.. 1960.

104, p.217-303.

[?] Comerford Jr.L.P., Edmunds C.C. Solutions of equations in free gmups.

Walter de Gruyter, Berlin, Xem York. 1989, pp. 347-356

[3] Fitting H. Die Theorie der Autornorphisrnenringe abelscher Gruppen und

ihr unalogon bei nichtcommutativen Gruppen. Math. .-\nn. vol. 107 (1933).

pp 514-543.

[A] Frohlich A. Distributively generated near-rings. Proc. Lond. Math. Soc. 8

(1958), pp. 76-108.

[5] Frohlich A. On gmups over a.d.g. neur-rings. Quart. J . Math. (Oxford)

(2) II (1960), pp. 193-225.

[6] Frohlich -4. Non-abelian hornological algebras. I. Denved functors and

satellites. Proc. Lond. Math. Soc. II (1961). pp. 239-275.

(71 Frohlich A. Non-abelian homological algebms. II. Varïeties. Proc. Lond.

Math. Soc. 12 (1962), pp. 1-25.

[S] Frohlich A. Non-abelian homological aige bras. III. The funct ors EXT and

TOR. Proc. Lond. Math. Soc. 12 (1962), pp. 739-768.

[9] Ghys E., P. de la Harpe Sur les groupes ht~perbolzqves d'apres Mzkhuel

Gromov. Progress in hlathematics, v. 83, 1990.

[IO] Gildenhuys D.? Kharlampovich O., Myasnikov A. CSA -groups and sep-

arated free constnictions. Bulletin of the hustralian Math. Soc. 1995. v.

32, 1, p.63-84.

[Il] Grigorchuk R.I., Kurchanov P.F. On quadratic equutions in free qroups.

Contemp. Math., 1992, v. 131, n. 1. pp. 159-171

(121 Grunewald F., Segal D. Reflections on the classification of torsion- free

nilpotent groups Group Theory: essays for Philip Hall. 1954, p. 121-1%.

(131 Hall P. Nilpo tent groups. Canad. Math. Congress, Edmonton. 1957.

[l4] Heerema X. Sums of nonnul endonrorphisms. Trans. Amer. Math. Soc.

-1957.-v.84.-pp. 137-147

[l5] Kargapolov hI.1.' Merzljakov Ju.1. Fundamentals O/ the Theory of Groups.

[16] Kharlampovich O., Myasnikov A. Turski 's problem about the eleinenta*y

theory of free groups has a positi-ue solution. ER;\-ALIS'9S.

[17] Lyndon R., Paul E. Schupp Combinatonal Group T h e o y .

(181 Lyndon R. Grovps with parametric eqonents . Trans. Amer. Math. Soc.

1960.-V. 96.-p. 518-533.

[19] Magnus W.' Karras A., Solitar D. Combinatorial group theoq. Inter-

science publ., 1966, pp. 1-44.

[20] Makanin G.S. Equations in a free group. Izv. .\kacl. Nauk SSSR. Ser. Mat..

1982. v. 46. pp. 1199-19'73

[XI blal'cev A.I. vilp pote nt torsion-lree groups. Izvestia Akad. Nauk CSSR.

Ser.Mat. 13 (1949). p.9-32.

[22] hleldrum J.D.P. Near-rings and ttheir links luith groups. 1985. Pitman.

London.

(231 Myasnikov A.G., Remeslennikov V. N. Eqonential groups 1 : Fo~undations

of the theory and tensor completions. Siberian Math. J.? 1993. v.5..

(-41 Myasnikov -1.G.' Remeslennikov V.N. Eqonential groups 2: Ertensions

of centralixrs und tensor cornpletion of CSA-gmups. Int. Journal of Al-

gebra and Cornputation, 1996. v.6. 3.6,~. 687-711.

[25] Myasnikov .A. Structure of models and criteria of decidability of completr

theones of finite-dimensiund algebras. Izvest. Akademii Xayk CSSR. ser.

math., 1989, v.53,2, p.379-39'7.

[26] Neumann H . Necir-rings connected uiith free gtoups Proc. Intern. Confer..

Amsterdam II (1954), pp. 46-47.

[27] Neumann H. On uarieties of groups and their associated near-rings Math.

Z. 65 (1956), pp. 36-69

[BI Pilz G. Near-rings. 'Jorth-Holland / American Elsevier. Amsterdam.

1983.

[29] Raz borov A.A. On sgstems of equations in free groups. Math. CSSR Izves-

tia. Ser. Mat., 1985, v. pp. 115-162

[30] Rips E.. Sela Z . Cunonical representatives und equations in hyperbola'c

groups. 1991

[3L] Robinson DA. Swns of n o n a l semi-endomorphisms. Amer. Math.

XIonth. -1963.-v. 70.-pp. 537-539.

i3-] CVarfield R.B. Nzlpotent groups. Lect. 'iotes in Math.. 513. Springer-

Verlag, Berlin. heidelberg, New York. 1976.