Patrick Dehornoy - Braids and Self Distributivity

8/8/2019 Patrick Dehornoy - Braids and Self Distributivity

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Contents

Preface vii

Part A: Ordering the Braids

I. Braids vs. Self-Distributive Systems 1

I.1 Braid Groups . . . . . . . . . . . . . . . . . . . . . . . 4I.2 Braid Colourings . . . . . . . . . . . . . . . . . . . . . . 13I.3 A Self-Distributive Operation on Braids . . . . . . . . . . . . 26I.4 Extended Braids . . . . . . . . . . . . . . . . . . . . . . 35I.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

II. Word Reversing 47

II.1 Complemented Presentations . . . . . . . . . . . . . . . . 48II.2 Coherent Complements . . . . . . . . . . . . . . . . . . . 58II.3 Garside Elements . . . . . . . . . . . . . . . . . . . . . 68II.4 The Case of Braids . . . . . . . . . . . . . . . . . . . . . 78II.5 Double Reversing . . . . . . . . . . . . . . . . . . . . . 87II.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 94

III. The Braid Order 97

III.1 More about Braid Colourings . . . . . . . . . . . . . . . . 98III.2 The Linear Ordering of Braids . . . . . . . . . . . . . . 106III.3 Handle Reduction . . . . . . . . . . . . . . . . . . . . 115III.4 Alternative Definitions . . . . . . . . . . . . . . . . . . 127III.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 137


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IV. The Order on Positive Braids 141

IV.1 The Case of Three Strands . . . . . . . . . . . . . . . . 142IV.2 The General Case . . . . . . . . . . . . . . . . . . . . 150

IV.3 Applications . . . . . . . . . . . . . . . . . . . . . . 165IV.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 170IV.Appendix: Rank Tables . . . . . . . . . . . . . . . . . . . 172

Part B: Free LD-systems

V. Orders on Free LD-systems 175

V.1 Free Systems . . . . . . . . . . . . . . . . . . . . . . 178V.2 The Absorption Property . . . . . . . . . . . . . . . . . 186V.3 The Confluence Property . . . . . . . . . . . . . . . . . 193V.4 The Comparison Property . . . . . . . . . . . . . . . . 200V.5 Canonical Orders . . . . . . . . . . . . . . . . . . . . 209V.6 Applications . . . . . . . . . . . . . . . . . . . . . . . 216

V.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 231

VI. Normal Forms 235

VI.1 Terms as Trees . . . . . . . . . . . . . . . . . . . . . 236VI.2 The Cuts of a Term . . . . . . . . . . . . . . . . . . . 242VI.3 The -Normal Form . . . . . . . . . . . . . . . . . . . 250VI.4 The Right Normal Form . . . . . . . . . . . . . . . . . 256VI.5 Applications . . . . . . . . . . . . . . . . . . . . . . 271VI.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 276VI.Appendix: The First Codes . . . . . . . . . . . . . . . . . 280

VII. The Geometry Monoid 285

VII.1 Left Self-Distributivity Operators . . . . . . . . . . . . . 286VII.2 Relations in the Geometry Monoid . . . . . . . . . . . . 301VII.3 Syntactic Proofs . . . . . . . . . . . . . . . . . . . . 310VII.4 Cuts and Collapsing . . . . . . . . . . . . . . . . . . . 322VII.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 329

VIII. The Group of Left Self-Distributivity 331

VIII.1 The Group GLD and the Monoid MLD . . . . . . . . . . . 332VIII.2 The Blueprint of a Term . . . . . . . . . . . . . . . . 343VIII.3 Order Properties in GLD . . . . . . . . . . . . . . . . . 356VIII.4 Parabolic Subgroups . . . . . . . . . . . . . . . . . . 367VIII.5 Simple Elements in MLD . . . . . . . . . . . . . . . . . 371VIII.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 383


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IX. Progressive Expansions 385

IX.1 The Polish Algorithm . . . . . . . . . . . . . . . . . . 386IX.2 The Content of a Term . . . . . . . . . . . . . . . . . . 395

IX.3 Perfect Terms . . . . . . . . . . . . . . . . . . . . . . 401IX.4 Convergence Results . . . . . . . . . . . . . . . . . . . 415IX.5 Effective Decompositions . . . . . . . . . . . . . . . . . 421IX.6 The Embedding Conjecture . . . . . . . . . . . . . . . . 428IX.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 436IX.Appendix: Other Algebraic Identities . . . . . . . . . . . . . 439

Part C: Other LD-Systems

X. More LD-Systems 443

X.1 The Laver Tables . . . . . . . . . . . . . . . . . . . . 446X.2 Finite Monogenic LD-systems . . . . . . . . . . . . . . . 457X.3 Multi-LD-Systems . . . . . . . . . . . . . . . . . . . . 467

X.4 Idempotent LD-systems . . . . . . . . . . . . . . . . . . 472X.5 Two-Sided Self-Distributivity . . . . . . . . . . . . . . . 476X.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 480X.App endix: The First Laver Tables . . . . . . . . . . . . . . 485

XI. LD-Monoids 489

XI.1 LD-Monoids . . . . . . . . . . . . . . . . . . . . . . 490XI.2 Completion of an LD-System . . . . . . . . . . . . . . . 498XI.3 Free LD-Monoids . . . . . . . . . . . . . . . . . . . . 511XI.4 Orders on Free LD-Monoids . . . . . . . . . . . . . . . 517XI.5 Extended Braids . . . . . . . . . . . . . . . . . . . . 527XI.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 534

XII. Elementary Embeddings 537XII.1 Large Cardinals . . . . . . . . . . . . . . . . . . . . 538XII.2 Elementary Embeddings . . . . . . . . . . . . . . . . . 546XII.3 Operations on Elementary Embeddings . . . . . . . . . . 553XII.4 Finite Quotients . . . . . . . . . . . . . . . . . . . . 559XII.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 568

XIII. More about the Laver Tables 571

XIII.1 A Dictionary . . . . . . . . . . . . . . . . . . . . . 572XIII.2 Computation of Rows . . . . . . . . . . . . . . . . . . 577XIII.3 Fast Growing Functions . . . . . . . . . . . . . . . . . 583XIII.4 Lower Bounds . . . . . . . . . . . . . . . . . . . . . 591XIII.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 598


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Bibliography 603

List of Symbols 615

Index 619


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Preface

The aim of this book is to present recently discovered connections betweenArtins braid groups Bn and left self-distributive systems (also called LD-systems), which are sets equipped with a binary operation satisfying the leftself-distributivity identity

x(yz) = (xy)(xz). (LD)

Such connections appeared in set theory in the 1980s and led to the discoveryin 1991 of a left invariant linear order on the braid groups.

Braids and self-distributivity have been studied for a long time. Braid groupswere introduced in the 1930s by E. Artin, and they have played an increas-ing role in mathematics in view of their connection with many fields, such asknot theory, algebraic combinatorics, quantum groups and the YangBaxterequation, etc. LD-systems have also been considered for several decades: earlyexamples are mentioned in the beginning of the 20th century, and the firstgeneral results can be traced back to Belousov in the 1960s. The existence ofa connection between braids and left self-distributivity has been observed andused in low dimensional topology for more than twenty years, in particular inwork by Joyce, Brieskorn, Kauffman and their students. Brieskorn mentionsthat the connection is already implicit in (Hurwitz 1891).

The results we shall concentrate on here rely on a new approach developedin the late 1980s and originating from set theory. Most of the examples ofself-distributive operations known at the time were connected with conjugacyin a group, and what set theory provided was a new example of a completelydifferent typetogether with the hint that something deep was hidden there,and the frustrating situation that the very existence of the example was anunprovable statement. Developing alternative constructions without relyingon an unprovable statement has led to a new geometrical approach to self-distributive algebra, which has made the connection with braids more striking,and has led to a number of results about left self-distributivity, in particularmany new examples and a complete description of free LD-systems; it has also


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led to new results about braids, the most promising so far being the existenceof a natural linear order on braids.

Our story began around 1983 when the main challenge of set theory was to

establish a connection between large cardinal axioms and Projective Determi-nacy, a structural statement that describes the Lusin hierarchy on the real line.Most large cardinal axioms involve elementary embeddings, which are a kindof endomorphism relevant in the context of set theory, and it was part of thefolklore that certain elementary embeddings can be equipped with a left self-distributive operation, resulting in an intricate calculus of iterates. A proof ofProjective Determinacy using large cardinals, namely Woodin cardinals, wasgiven by Martin and Steel in 1985, and this proof, as well as a subsequentalternative proof by Woodin, requires much more than computing iterates ofelementary embeddings. However, two results of the time (1986) are the au-thors observation that the existence of the left self-distributive operation onelementary embeddings is sufficient to prove Analytic Determinacy, a nontrivialfragment of Projective Determinacy involving the first two levels of the Lusinhierarchy, and Lavers theorem (which appeared only later) that left division

in the LD-systems of elementary embeddings has no cycle. The former resultshows that the self-distributive operation is nontrivial in a strong sense; thelatter shows that it is quite different from the conjugacy in a group.

Motivated by the previous observations, R. Laver and the author undertook asystematic study both of the LD-systems of elementary embeddings and of freeLD-systems. Contrary to related free systems, such as Joyces free quandlesor Fenn and Rourkes free racks, the free LD-systems had not received anyattention so far, probably because examples were missing. The most significantresult obtained during this period was the existence of a left invariant linearorder on free LD-systems of rank 1 and, as a corollary, a solution to the wordproblem of Identity (LD), under the hypothesis that there existed an LD-system in which left division has no cycle; this was proved by R. Laver and theauthor independently in the Spring of 1989.

The above results created a strange situation: the existence of a left invari-ant order on free LD-systems, or the decidability of the word problem of ( LD),are effective, finitistic properties; yet, their proof required the existence of anLD-system with a certain property, and the only known examples of such LD-systems were the LD-systems of elementary embeddings. Now, an inevitableconsequence of Godels incompleteness theorem is that the existence of an ele-mentary embedding of the required type is an unprovable statement, actuallya higher infinity axiom. Hence the existence of an LD-system of elementaryembeddings is unprovable. Therefore, the logical status of the previous resultswas unclear, and new developments were needed, either to construct alterna-tive examples of LD-systems resorting to no set theoretical axiom, or to provethat such axioms were needed, as is known to be in the case of ProjectiveDeterminacy.

Here braids came into the picture, indirectly at first. The main problem was


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to study free LD-systems. As no concrete realization was available, the onlypossible approach was a syntactic approach. The main idea here has been tointroduce a certain group GLD which captures some geometrical properties of

the identity (LD

). The groupGLD

happens to be an extension of the unionB

of all Artins braid groups Bn, and it has been investigated using algebraictools analogous to the ones Garside used for braid groups. At the end of 1991,this study led to a proofwithout any set theoretical hypothesisthat leftdivision in free LD-systems has no cycle and, as a natural by-product, to theconstruction of a left self-distributive operation and of a left invariant linearorder on braids.

Further results both about braid ordering and about LD-systems were thenproved by several authors. First, once the explicit definition of the above-mentioned self-distributive operation on braids became available, a shorterproof for existence of the braid ordering could be given, as was noted by Laruein 1992. In 1993, Laver proved that the restriction of the order to n-strandpositive braids is a well-ordering. In 1995, we derived from the braid orderinga new efficient method for recognizing braid isotopy. In 1997, Fenn, Greene,

Rolfsen, Rourke, and Wiest reconstructed the braid order by purely topologicalmeans, and extended the method to the mapping class group of any surfacewith a nonempty boundary. More recently, Short and Wiest, building on asuggestion by Thurston and work by Nielsen, gave another definition involvinghyperbolic geometry and an action of braids on the real line.

As for self-distributive algebra, i.e., the study of LD-systems, the main de-velopments involve the free LD-systems, the LD-systems of elementary embed-dings, and some finite LD-systems which we call the Laver tables. The resultsabout free LD-systems consist mainly in the construction of several normalforms by Laver and the author (around 1992), and in a deepening of our under-standing of the group GLD (recent work). About the LD-systems of elementaryembeddings, a number of results were proved by Laver, Dougherty, and Jechbetween 1988 and 1995 in connection with the computation of the so-calledcritical ordinals and attempts to construct finitistic counterparts that do notrely on large cardinal axioms. The Laver tables were introduced by Laver in the1980s as finite quotients of the LD-systems of elementary embeddings. They arefascinating objects with a formidable combinatorial complexity. Many resultsabout them were proved in the 1990s by Drapal. A remarkable point is that,contrary to the existence of an LD-system with an acyclic left division, someresults about the Laver tables, proved using elementary embeddings, have notyet received alternative combinatorial proofs, and, therefore, they still dependon an unprovable logical axiom.

Looking at the current picture, we see that set theory is not involved in anyof the braid constructions, and one may feel that its involvement in the resultsabout the Laver tables simply implies that our understanding of these objectsis far from complete. However, we would argue that both the braid order andthe Laver tables are, and are to remain, applications of set theory: if the latter


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had not clearly shown the way, it is more than likely that most of the results weshall present in this monograph would not have been discovered yet. Similarly,the existence of the braid order can now be established quickly using Larues

short proof that left division associated with the left self-distributive operationon braids has no cycle: yet, it is not clear that this operation would have beendiscovered without a study of the group GLD.

About this book

The current text proposes a first synthesis of this area of research. Our exposi-tion is self-contained, and there are no prerequisites. This leads us to establisha number of basic results, about braids, self-distributive algebra, and, to someextent, set theory. However, the text is not a comprehensive course about thesesubjects, as we have selected only those results that are needed for our specific,mainly geometryoriented, approach.

The text is divided into three parts, devoted to the braid order, to free LD-

systems, and to general LD-systems respectively. This order does not follow thechronology of development, but it gives the braid applications first in order tomotivate the study of free LD-systems that comes next. Set theory is postponedto the end, when we cannot avoid it any longer. The three parts are ratherindependent, and it is possible to begin with any of them, at least for a firstglance.

Let us give a preview of each part of the book. The aim of Part A is to constructthe linear order of braids and establish its main properties. The point is theexistence of an action of braids on powers of LD-systems: if S is a given LD-system, and b is an n-strand braid, then, for each element a in Sn, we candefine a new element a b by seeing a as a set of colours put on the top of band a b as the resulting set of colours at the bottom ofb when the colours flowdown b according to the rule

x y

xy x

.

When one uses classical examples of LD-systems, such as the conjugacy in agroup or the barycentric mean, one deduces well-known results, such as therepresentation of braids inside automorphisms of a free group, and the Buraurepresentation. New results appear when we use non-classical examples of LD-systems. In particular, a linear ordering of braids appears naturally when weuse a linearly ordered LD-system (S,


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linear order on S satisfies a < ab for all a, b, and the main technical result thatmakes the construction possible is that the free LD-system of rank 1 admitssuch a linear ordering. Technically, we avoid using abstract free LD-systems

here and we resort instead to a self-distributive operation defined on braids.Part A comprises four chapters. In Chapter I, we give an introductionto braids, and we present quickly (and somewhat informally) the connectionsbetween braids and self-distributivity, namely the above-mentioned action ofbraids on LD-systems, and the existence of a left self-distributive operation(called exponentiation) on braids. We also introduce what we call extendedbraids.

Chapter II is a preparatory chapter in which we develop a specific combi-natorial method for studying those groups and monoids with a presentation ofa particular syntactic form. We show that the braid groups are eligible, anddeduce a number of classical properties of these groups. The method is usedin Chapter VIII again.

Chapter III is devoted to the linear order of braids. Using the word reversingtechnique of Chapter II, we extend the action of braids on powers of LD-systems

defined in Chapter I to more general LD-systems. Admitting a general resultabout monogenic LD-systems that will be proved in Chapter V, we deducethe existence of a left invariant order on braids. We also mention alternativedefinitions of this order.

In Chapter IV, we study the restriction of the braid order to positive braids,i.e., to those braids that can be expressed without using the inverses 1

iof

the standard braid group generators. The main result is Lavers theorem thatthe restriction of the order to n-strand positive braids is a well-ordering. Herewe present Burckels construction, which computes the order type effectivelyand provides a new normal form for positive braids.

Part B is a general study of free LD-systems. Should the associativity identityreplace left self-distributivity, the free systems would be the free semigroups,hence rather simple objects. Free LD-systems are actually much more compli-cated, yet they share many properties with free semigroups, and, in particular,the property that the free system of rank 1 is equipped with a unique linearorder such that a < ab always holds: in the case of associativity, the free semi-group of rank 1 is the set of positive integers, and the corresponding linearorder is the usual order of the integers. The core of our study consists in in-vestigating LD-equivalence of terms. The latter are abstract expressions (orwords) involving variables, one binary operator, and parentheses, and we saythat two terms t, t are LD-equivalent if we can transform t into t by applyingIdentity (LD) once or several times. If we think of associativity again, apply-ing the identity to a term amounts to changing the place of parentheses, sothat every equivalence class is finite, and we can select a unique representa-tive easily, for instance by pushing all parentheses to the right. The case ofLD-equivalence is much more complicated: in general, the equivalence class of


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a term is infinite, and finding distinguished representatives is a serious task.The main point in the approach we develop here is to put the emphasis ongeometrical features*. We introduce a certain monoid GLD consisting of partial

mappings on terms, and a related groupGLD

, so that the LD-equivalence classof a term t becomes its orbit under some partial action of GLD**. All resultsabout braids mentioned in Part A then originate from results on GLD: the ex-istence of the braid action on LD-systems comes from Artins braid group Bbeing a quotient of GLD; braid exponentiation originates from expressing in GLDa simple general property of left self-distributivity called absorption; the linearordering of braids comes from some linear preordering on GLD. The situationcan be summarized in the slogan

The geometry of left self-distributivity is a refinement of the

geometry of braids,

which is reminiscent of the well-known observation that the geometry of braidsis a refinement of the geometry of permutations: going from the symmetricgroup S to the braid group B amounts to completing a permutation with

some information about how the transpositions have been performed; similarly,going from B to GLD amounts to completing a braid with some additionalinformation about the names of the strands that have been braided, as thediagram below may suggest:

Braiding Applying left self-distributivity

x y z x y z x y z

y x z x y x z (xy) x z .

Part B is divided into five chapters. In Chapter V, we introduce free LD-systems and develop a convenient framework of finite trees. We prove theComparison Property, which is the missing fragment in the construction of the

braid order in Chapter III, and the existence of left invariant linear orders onfree LD-systems, which implies the decidability of the word problem.

In Chapter VI, we prove unique normal form results for LD-equivalence, i.e.,we construct families of distinguished terms so that every term is LD-equivalentto exactly one distinguished term. This requires developing a precise analysisof the geometry of terms. Various applications are given, in particular aboutbraids and their exponentiation.

* R. Laver has developed an alternative approach, briefly mentioned at theend of Chapter VI; neither approach is subsumed in the other, as both provestatements seemingly inaccessible to the other.

** A similar approach can be developed for every identity: in the case of asso-ciativity, the corresponding group is Richard Thompsons group F.


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In Chapter VII, we introduce the monoid GLD by considering partial op-erators acting on terms and specifying how and where left self-distributivityis applied. We exhibit a family of relations, called LD-relations, which hold

in GLD

, reminiscent of those relations involved in the MacLaneStasheff pen-tagon in the case of associativity. We check that most of the results about freeLD-systems established in Chapter V can be deduced from LD-relations.

In Chapter VIII, we introduce the group GLD for which LD-relations yielda presentation. We describe the connection between GLD and GLD, and we con-struct counterparts to the braid exponentiation and to the braid order in GLD.In particular, we give a purely syntactical proof of the fundamental result thatleft division in a free LD-system admits no cycle.

In Chapter IX, we deepen our study of the group GLD and of the associatedpositive monoid MLD. By refining the results of Chapter VI, we prove par-tial results about the Polish Algorithm and the Embedding Conjecture: ThePolish Algorithm is a natural syntactic method for deciding LD-equivalence ofterms, and its termination is one of the most puzzling open questions of thesubject. The Embedding Conjecture claims that the monoid MLD embeds in

the group GLD, a counterpart to Garsides result that the braid monoid B+n em-beds in the group Bn. This conjecture is equivalent to a number of propertiesof left self-distributivity.

Part C contains further developments about LD-systems. We did not try tobe exhaustive, and we concentrated on some special families of LD-systems,in particular the Laver tables and the LD-systems of elementary embeddings.The Laver tables form an infinite family A0, A1, . . . of finite monogenic LD-systems. The LD-system An has 2

n elements, and it is the unique LD-systemwith domain {1, 2, . . . , 2n} satisfying a1 = a+1 mod 2n for every a; projectionmodulo 2n defines a surjective homomorphism of An+1 onto An for every n, sothe family (An)n is an inverse system. On the other hand, ranks are specialsets with the weird property that every mapping ofR into itself can be seen asan element ofR, and elementary embeddings are defined as nontrivial injectivemappings that preserve every notion that is definable from the membershiprelation . Let us say that a rank R is self-similar if there exists an elementaryembedding ofR into itself which is not the identity. The point is that, if i and jare elementary embeddings of some self-similar rank R into itself, then, as i ap-plies to every element ofR and j can be seen as an element ofR, we can apply ito j, thus obtaining a new elementary embedding denoted i[j]. By construc-tion, the bracket operation on elementary embeddings is left self-distributive,and the main result is that the Laver tables An are natural quotients of theLD-systems of elementary embeddings thus obtained. This connection resultsin a dictionary between elementary embeddings in set theory and values in theLaver tables. In particular, the LaverSteel theorem, a deep well-foundednessresult about elementary embeddings, translates into the result that the numberof values occurring in the first row of An tends to infinity with n. The puzzling


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point is that no direct proof of the latter combinatorial statement has beendiscovered so far. So, as the existence of a self-similar rank is an unprovableaxiom, the logical status of this statement remains open.

Part C consists of four chapters. In Chapter X, we introduce the Lavertables, we present Drapals classification of finite monogenic LD-systems, andwe mention other families of LD-systems, such as idempotent and two-sidedself-distributive LD-systems.

In Chapter XI, we study LD-monoids, which are LD-systems equipped withan additional compatible associative operation. We solve the natural problemof completing a given LD-system into an LD-monoid, and we give a completedescription of free LD-monoids. We also study the LD-monoid of extendedbraids introduced in Chapter I.

In Chapter XII, we describe the LD-systems of elementary embeddings.Here, we give a short self-contained introduction to the relevant facts neededfrom set theory, and show how self-distributivity naturally appears. Using awell-foundedness argumentthe core of set theorywe establish the LaverSteel theorem, which is crucial in further applications.

In Chapter XIII, we return to the Laver tables and we give a speculativeconclusion to the book. We construct the above mentioned dictionary betweenelementary embeddings and Laver tables. Building on the unprovable hypoth-esis postulating the existence of a self-similar rank, we deduce results aboutthe values in An, and we show why a direct combinatorial proof of the latterresults has to be very complicated.

The methods used here are mostly algebraic and combinatorial in nature. Theemphasis is put on words, rather than on the elements of a monoid, a group,or an LD-system they represent. Only such an approach allows us to studyfine features which are not visible in the monoid, the group or the LD-systembecause the relations they involve are not compatible with the congruencedefining the considered monoid, group, or LD-system. For instance, the notionof braid word reversing introduced in Chapter II is a refinement of the standardnotion of braid word equivalence, and both induce equality of braids. However,using the former relation rather than the latter is crucial for defining the actionof braids on non-classical LD-systems and deriving applications such as thebraid order. This perhaps explains why such an action had not been consideredbefore.

Exercises appear at the end of most sections, usually with the aim of men-tioning in a short way further results (a lot of them correspond to unpublishedmaterial).

Historical remarks and proper credits are given in the notes at the endof each chapter. One precision is in order. The current text is deliberatelycentered on the authors approach to the subjectwith the noticeable exceptionof Chapters IV, X, XII, and XIII, which rely on work by Burckel, Laver, Drapal,and Dougherty. However, we by no means claim that all unattributed results


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are ours: a number of them is part of the folklore, and there is no doubt thatRichard Laver has known a lot of them for many years.

It is a pleasure to express my thanks to P. Ageron, G. Basset, S. Burckel,

A. Drapal, A. Kanamori, R. Laver, B. Leclerc, J. Lescot, A. Sossinsky, andB. Wiest for valuable discussions, comments and corrections. I owe specialthanks to T. Kepka, who prepared the historical notes at the end of Chapter X,to M. Picantin, who found uncountably many mistakes in the manuscript, andto C. Kassel, who suggested a great number of improvements.

December 1999Patrick Dehornoy


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I.1 Braid Groups . . . . . . . . . . . . . . . . . . . . 4

I.2 Braid Colourings . . . . . . . . . . . . . . . . . . . 13I.3 A Self-Distributive Operation on Braids . . . . . . . . . 26I.4 Extended Braids . . . . . . . . . . . . . . . . . . . 35I.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . 44

This chapter gives a quick introduction to our main subject, namely the con-nection between braids and self-distributive operations. Our presentation isbased on the concrete idea of colouring the strands of a braid, and left self-distributivity arises as a natural compatibility condition. At this early stage,some of the constructions may look artificial or strange: it will be one of theaims of the subsequent chapters, in particular in Part B of this book, to explainthem and hopefully make all of them natural.

Section 1 contains an elementary introduction to Artins braid groups. Weestablish the standard presentation of the braid group, a basis for further al-gebraic developments. In Section 2, we describe an action of braids on thosealgebraic systems that involve a self-distributive operation. We review theclassical examples of such systems, and mention the braid properties obtainedusing the action. We also describe a non-classical example involving injec-tions. In Section 3, we show how a self-distributive operation, called braidexponentiation, can be constructed inside Artins braid group B by usingbraid colourings. We present some related self-distributive systems, and weprove that left division associated with braid exponentiation has no cycle, acrucial result for subsequent order properties. Finally, in Section 4, we con-sider LD-monoids, which are structures involving both a left self-distributiveoperation and a compatible associative product. LD-monoids naturally appearwhen colouring braid diagrams. We construct such a structure on a partialtopological completion of the braid group B corresponding to the intuitiveidea of strands vanishing at infinity.


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4 Chapter I: Braids vs. Self-Distributive Systems

1. Braid Groups

There are many ways to introduce Artins braid groups. Here, we choose themost elementary approach, namely starting from the intuitive idea of a concretebraid.

Geometric braids

Definition 1.1. (geometric braid) Assume that f is a permutation of 1,. . . , n. An n-strand geometric braid associated with f is defined to be theunion ofn disjoint polygonal arcs 1, . . . , n in R

2[0, 1] such that i connects(f(i), 0, 0) to (i, 0, 1) for each i and, for every z0 in [0, 1], the intersection of iwith the plane z = z0 consists of exactly one point. If is a geometric braidassociated with f, we say that f is the permutation of , and we denote it

by perm().

In Figure 1.1, we display an example of a 4-strand geometric braid. In allpictures, the z-axis will correspond to the vertical direction of the figure, witha downward orientation. The last condition in the definition implies that thez-coordinate increases monotonically along each arc.

Figure 1.1. A 4-strand geometric braid

Definition 1.2. (product) (Figure 1.2) Assume that 1 and 2 are n-strandgeometric braids. The product (or composition) 12 is defined to be the geo-metric braid obtained by stacking 1 over 2 and shrinking the result so thatit lies in R2 [0, 1]so, formally, 12 is c(1 t(2)), where t is translationby (0, 0, 1) and c is the affinity that multiplies the z-coordinate by 1/2.

1 212

=

Figure 1.2. Product of geometric braids


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Section I.1: Braid Groups 5

Lemma 1.3. The relation = 12 implies perm() = perm(1) perm(2).

The result is obvious. As it stands, the product of braids is not associative.But it becomes so provided we allow the strands to be stretched or shrunk.More generally, we shall consider as equivalent two geometric braids such thatone is obtained from the other by moving strands but not allowing them tocross one another.

Definition 1.4. (isotopy) (Figure 1.3) Assume that , are geometricbraids. We say that is 1-isotopic to if is obtained from by replacingtwo consecutive segments [AB], [BC] with the segment [AC], or vice versa,provided that the convex hull of ABC does not intersect . We say that isisotopic to if there exists a finite sequence 0 = , 1, . . . , m =

such thati is 1-isotopic to i1 for every i.

A A

B B

C C

Figure 1.3. Isotopy

By construction, isotopy is an equivalence relation, and it preserves the numberof strands and the permutation.

Definition 1.5. (braid) An n-strand braid is an isotopy class of geometricn-strand braids. The set of all n-strand braids is denoted by Bn.

Lemma 1.6. Assume that h is a piecewise affine bijection of [0, 1] into itself.

Let h be the function of R2 [0, 1] that maps (x,y,z) to (x,y,h(z)). Thenevery geometric braid is isotopic to its image under h.Proof. By definition, isotopy is a transitive relation, so we may assume thatthe graph of h consists of only two segments, and, therefore, it lies eitherabove or below the diagonal. Assume the first. Then h lowers each pointof R2 (0, 1). Let be a geometric braid. We define inductively an isotopy

of onto h(): assuming that [AB] and [BC] are consecutive segments in , wefirst replace [BC] with [B

h(B)][

h(B)C], and, then, we replace [AB][B

h(B)]

with [Ah(B)]. The possible problem is that some arc of may intersect thetriangles [ABh(B)] or [Bh(B)C]. A solution consists in ordering the segmentsof as follows: we say that is below if the projections of and on theplane z = 0 intersect at some point (x,y, 0) and, letting (x,y,z) and (x,y,z)be the liftings of (x,y, 0) in and respectively, z > z holdsaccording to


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our graphical conventions, z = 0 lies above z = 1. This relation need not be anordering in general (cycles may exist), but it becomes an ordering provided we

possibly partition the segments into smaller pieces. Then no problem will ariseprovided we enumerate the segments so as to begin with the lowest ones.

Proposition 1.7. (braid group) The product of geometric braids inducesa well-defined product on Bn, which gives Bn the structure of a group. Themapping perm induces a surjective homomorphism of Bn onto the symmetricgroup Sn.

Proof. By construction, if 1 is isotopic to 1, 12 is isotopic to 12 for

every 2. So isotopy is compatible with multiplication on the right, and, sim-ilarly, it is compatible on the left. Hence, the product on Bn is well-defined.Applying Lemma 1.6 to the function h defined by

h(z) = 2z for 0 z 1/4,z + 1/4 for 1/4 z 1/2,z/2 + 1/2 for 1/2 z 1

shows that (12)3 is isotopic to 1(23) for all 1, 2, 3. So, the producton Bn is associative. Next, define h by

h(z) =

2z for 0 z 1/2,1 for 1/2 z 1.

The existence of h shows that every geometric n-strand braid is isotopicto n, where n denotes the trivial geometric braid consisting of n verticalsegments (the current mapping h is not bijective, but the argument remainsthe same). Similarly, is isotopic to n, and the class of n is a unit for the

product of Bn.Next, let s denote the symmetry with respect to the plane z = 1/2. The

braids s() and s() are invariant under s, and we claim that every n-strandbraid that is invariant under s is isotopic to n. This will prove that the classof s() is the inverse of the class of in Bn. So assume that is a symmetricgeometric braid consisting of n arcs 1, . . . , n. Let i denote the convexhull of i. Up to using an isotopy to slightly move some segments, we mayassume that the union of all intersections ij consists of a finite number p ofvertical segments. Let us call this number p the intersection number of . Forp = 0, we can use an isotopy to transform each arc i into a vertical segment,hence is isotopic to n. Otherwise, let be an intersection segment withminimal length. Assume that the ends of lie on i and in the interior of j .By construction, the part of i that is bounded by and contains no pointwith z-coordinate 0 or 1 cannot intersect . Hence we can transform into anisotopic symmetric braid with smaller intersection number by replacing slightlymore than the central portion of i determined by with a vertical segment(Figure 1.4).


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Finally, the permutations of isotopic braids are equal, and Lemma 1.3 tellsus that perm is a homomorphism. This homomorphism is surjective, as trans-

positions generate the symmetric group, and constructing a geometric braidwhose permutation is a given transposition is trivial.

ij

ij

Figure 1.4. Diminishing the intersection number

For each n, we shall denote by 1 the unit ofBn, which we have seen is the classof the trivial geometric n-strand braid n.

Presentation of Bn

The aim of this subsection is to prove:

Proposition 1.8. (presentation) The group Bn admits the presentation

1, . . . , n1 ; ij = ji for |i j| 2, iji = jij for |i j| = 1.

To this end, we consider the orthogonal projections of geometric braids on theplane y = 0. We shall refer to this projection as the y-projection. The idea isto consider geometric braids with a standardized y-projection that can be fullydescribed by a word over a finite alphabet, and, then, to characterize thosewords that code for isotopic braids.

Definition 1.9. (regular) A geometric braid is said to be regular if its y-projection has no triple point, it has only finitely many double points, thez-coordinates of the double points are pairwise distinct, and the double pointsare not the projection of the end of a segment.

Lemma 1.10. Every geometric braid is isotopic to a regular one.

Proof. Regular braids form a dense set with respect to the topology on geomet-ric braids induced by the ambient topology of R3, while the set of all geometricbraids that are isotopic to a given geometric braid is open. Hence these setsintersect.


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Assume that d is the y-projection of a regular braid . Each doublepoint (x, 0, z) of d is the y-projection of two points (x, y1, z) and (x, y2, z)

of that belong respectively to two segments [A1B1] and [A2B2]. We say that[A1B1] lies in front of [A2B2] if y1 > y2 holds.

Definition 1.11. (braid diagram, realization) A braid diagram is definedto be the y-projection of a regular braid, together with the specification, foreach double point, of which segment lies in front. If d is a braid diagram, thenormal realization of d is the geometric braid d obtained from d by replacing,for each double point M of d, a sufficiently short section [AC] of the segmentcontaining M and lying in front of the other with the union of two connectedsegments [AB], [BC] such that B lies in the plane y = 1 and M is its y-projection.

We represent a braid diagram with small interruptions on back segments, asin Figure 1.5. As it stands, the normal realization of a braid diagram is notunique, but it would be easy to make it so, the only requirement being thatthe y-projection of d be d and d be simple enough.

Figure 1.5. A braid diagram

Lemma 1.12. Assume that and are regular braids that project on thesame braid diagram. Then and are isotopic.

Proof. Letting d be the y-projection of and , we show that is isotopicto d using induction on the number of crossings in d. The positions of the crossings in a braid diagram can be coded by a word.Assume that d is a braid diagram, and [AB] is a segment in d that is shortenough to contain at most one double point. Assume zA < zB. We say that[AB] lies at position i ifi 1 points in the intersection of d with the horizontalline z = zA lie on the left of A, i.e., lie on the half-line x < xA.

Definition 1.13. (code, braid word) Assume that d is an n-strand braiddiagram. Let z1 < < z be the z-coordinates of the double points in d.The code of d is defined to be the word e1i1

ei

, where ik and ik + 1 are thepositions of the segments that cross on the line z = zk, and ek = +1 (resp. 1)


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Thus, if two geometric n-strand braids are coded by the same word, they areisotopic. In order to completely describe the isotopy classes of geometric braids,

it remains to recognize in which cases different braid words code for isotopicbraids. The following lemma gives the answer, and it completes the proof ofProposition 1.8 (presentation).

Lemma 1.16. Let be the least congruence on the monoid BWn such that

i j j i for |i j| 2, (1.1)

i j i j i j for |i j| = 1, (1.2)

hold, and we have i1i

1i i for every i. Then two words in BWn

code for isotopic geometric braids if and only if they are -equivalent.

Proof. On the one hand, the geometric braids Dn(ij) and Dn(ji) areisotopic for |i j| 2. Similarly, the braids Dn(ii+1i) and Dn(i+1ii+1)are isotopic, and so are Dn(i1i ), Dn(1i i) and Dn(). As braid isotopyis compatible with product and is the congruence generated by the previouspairs of words, w w implies that the geometric braids Dn(w) and Dn(w)are isotopic.

On the other hand, it suffices to show that, if is obtained from byreplacing two consecutive segments [AB], [BC] with [AC], then the codes of and are -equivalent. At the expense of possibly decomposing the isotopyinto several steps, we may assume that the projection of the triangle [ ABC]either intersects no other projection of a segment of , or it intersects one, orit intersects two which, in addition, intersect inside the triangle.

In the first case, the codes of and

are equal. In the second case, thereare two possibilities. Assume first that the projection of some segment of intersects the projections of [AC] and [BC]. These intersections contribute tothe codes of and . It follows from the definition of an isotopy that either lies in front both of [AC] and [BC], or it lies behind both of them, so thecontributions are the same letter 1i . However, the codes of and

neednot be equal, as the z-coordinates of the intersections associated with , say z0and z0, need not be equal. If no other intersection has z-coordinate between z0and z0, the codes coincide. Otherwise, some intersections that contribute to thecode before in will contribute after in , or vice versa. Now, this amountsto replacing some factor of the form ei

e

j with |i j| 2 by the corresponding

factor e

j ei . Owing to Relation (1.1), and, possibly, to k

1k

1k k ,

the codes of and are -equivalent.Assume now that the projection of some segment of intersects the pro-

jections of [AB] and [BC], but not the projection of [AB]. By possibly usingintermediate transformations of the previous type, we may assume that noother intersection occurs between the intersections of with [AB] and [BC].


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Then the latter contribute some factor ei ei in the code of, which is absent

in the code of . By definition, the codes of and are -equivalent.

Finally, assume that the projection of [ABC] contains the intersection ofthe projections of two segments , . There are several cases according to thevarious possible positions of the segments, but, as replacing [AB] and [BC]with [AC] is a legal isotopy, all of [AB], [BC] and [AC] lie in front of , orall lie behind , and the same holds for . Figure 1.7 displays the possiblecases and the corresponding codes, up to a symmetry that replaces every codewith its inverse. Using Relation (1.2) and k

1k

1k k , we see that the

codes of and are -equivalent.

ji vs. ij 1i i vs. i

1i vs.

ii+1ivs. i+1ii+1

1i i+1ivs. i+1i

1i+1

1i 1i+1i

vs. i+11i

1i+1

ii+11i

vs. 1i+1ii+1

1i+1ii+1ivs. ii+1

1i+11i i+1i

vs. i1i+1

i+1ii+11i

vs. ii+1

1i+11i

1i+1i

vs. 1i 1i+1

i1i+1

1i

vs. 1i+11i i+1

1i 1i+1i

vs. i+11i

1i+1

i+1i1i+1

1i

vs. 1i i+1

i+11i

1i+1

1i

vs. 1i 1i+1

Figure 1.7. Braid relations

By the previous result, the isotopy class of a braid is determined by the braid


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Section I.2: Braid Colourings 13

2. Braid Colourings

The first connection between braids and self-distributivity is the existence ofan action of braids on sequences of elements chosen in a set equipped with aleft self-distributive operation. The description of this action is natural whenwe use the intuition of colouring the strands of a braid.

Braid colourings

Assume that S is a fixed nonempty set, and d is an n-strand braid diagram.We use the elements of S to colour the strands of d. To this end, we attributeto each top end in d a colour from S, which we propagate along the strands of

the diagram. The idea is to recover information about the braid b representedby d by comparing the final sequence of colours with the initial sequence.

If the colours are propagated without change along the strands, the finalsequence is a permutation of the initial sequence, and the only datum about bwe recover is the permutation perm(b).

Things become more interesting when we allow colours to change at cross-ings. Let us begin with the case of positive braid diagrams. According toour principle of propagating the colours downwards from the top of the braiddiagram, it is natural to fix the rules of colouring so that the colours after acrossing (the new colours) are functions of the colours before the crossings(the old colours). Keeping the colours corresponds to the simplest function,namely identity. The next step is to assume that only one colour may change,say the colour of the front strand, and the new colour depends only on the two

colours of the strands that have crossed. This amounts to using a function ofS S into S, i.e., a binary operation on S, so that the rule for propagatingcolours is

a b

ab a .

This leads to the following formal definition. Here, and everywhere in thesequel, we adopt the convention that, when x represents a (finite or infinite)sequence, then the sucessive elements of x are denoted x1, x2, . . . .

Definition 2.1. (braid action) Assume that (S,

) is a binary system, i.e.,a set equipped with a binary operation. The (right) action of BW+n on S

n isdefined inductively by

a = a, a iw = (a1, . . . , ai ai+1, ai, ai+2, . . . , an) w. (2.1)


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As we are interested in braids rather than in braid words, we want the actionof (2.1) to induce a well-defined action of braids, i.e., the colourings have to be

invariant under braid relations.

Lemma 2.2. Assume that (S, ) is a binary system. The action ofBW+n on Sn

defined by (2.1) is always invariant under Relations (1.1); it is invariant underRelations (1.2) if and only if (S, ) satisfies the left self-distributivity identity

x (y z) = (x y) (x z). (LD)

Proof. Compare the diagrams:

a b c a b c

ab a c a bc b

ab ac a a(bc) a b

(ab)(ac) ab a , a(bc) ab a .

The lower colours on the left strand coincide for every initial choice of a, b, cif and only if the operation satisfies Identity (LD).

Let us now colour arbitrary braid diagrams, i.e., let us try to define an action ofarbitrary words on sequences of colours. We have to choose a rule for colouringnegative crossings. In order to make the definition flexible, let us first assume

that the set of colours S is equipped with two more binary operations, say and . We complete the definition with the rule

a b

a b ba .

This amounts to extending the action of braid words on Sn by

a 1i w = (a1, . . . , ai ai+1, ai+1 ai, ai+2, . . . , an) w. (2.2)

Lemma 2.3. Assume that (S,

,

,

) is a triple binary system. The rightaction of BWn on Sn defined by(2.1) and (2.2) is invariant under the relationsi

1i =

1i i = if and only if S satisfies the identities

x y = y, x (x y) = x (x y) = y.


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Proof. This follows from:a b a b

ab a a b ba

(ab)a a(ab), (ab)(ba) ab .

We see on the right strands that a b = b is required. Then looking at the leftstrands gives the relations between the operations and .

We are thus led to the following definitions:

Definition 2.4. (LD-system, LD-quasigroup) (i) An LD-system is de-fined to be a set equipped with a left self-distributive operation, i.e., a binaryoperation that satisfies Identity (LD).

(ii) An LD-quasigroup is defined to be an LD-system equipped with an ad-ditional operation that satisfies the identity

x (x y) = x (x y) = y. (2.3)

The following result summarizes the above observations. Although trivial, itwill be central for our purpose.

Proposition 2.5. (action) (i) Assume that (S, ) is an LD-system. Then, forevery n, Formula (2.1) defines an action of the braid monoid B+n on S

n.(ii) Assume that (S, , ) is an LD-quasigroup. Then, for every n, Formu-

las (2.1) and (2.2) define an action of the braid group Bn on Sn.

Remark 2.6. In the sequel, we shall use several notations for left self-distributive operations. When only one operation is considered, there is noproblem in using a dot notation, with the operator usually dropped. Whenseveral operations are considered simultaneously, it is often convenient to usea left exponentiation for the left self-distributive operation, thus writing xy forthe product ofx and y. In this framework, left self-distributivity is expressedby x(yz) =

xy(xz). For better readability, we use xy as an alternative to xy.

Convention 2.7. (parentheses) In all algebraic expressions, missing paren-theses are to be added on the right: thus, everywhere in this text, xyz standsfor x(yz), xyz stands for x(yz), etc.

Before going further, let us observe that the two operations of an LD-quasigroupdetermine each other. Let us recall that a binary system (S, ) is said to beleft cancellative if ab = ac implies b = c, and to be left divisible if, for all a, cin S, there exists at least one element b in S satisfying ab = c. In other words,


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if we denote by ada the left translation x ax, then (S, ) is left cancellativeif and only if all mappings ada are injective, and it is left divisible if and only

if all mappings ada are surjective.

Proposition 2.8. (second operation) (i) Assume that (Q, , ) is an LD-quasigroup. Then both (Q, ) and (Q, ) are left cancellative left divisibleLD-systems; the operations and are mutually left distributive, and eachone determines the other by:

a c = the unique b satisfying a b = c, (2.4)

a c = the unique b satisfying a b = c. (2.5)

(ii) Conversely, assume that (Q, ) is a left cancellative left divisible LD-system.Let be the operation defined by (2.4). Then (Q, , ) is an LD-quasigroup.

Proof. (i) By hypothesis, (Q, ) is an LD-system, i.e., the operation satisfiesIdentity (LD). If ada and ad

a denote the left translationsx ax and x ax

respectively, Formula (2.3) gives ada ad

a = ad

a ad

a = idQ for every a, and,therefore, it implies that every left translation of (Q, ) and (Q, ) is a bijection.Hence (Q, ) and (Q, ) are left cancellative and left divisible, and (2.4) and(2.5) hold.

We prove now that (Q, ) is an LD-system. Let a, b, c be arbitrary elementsof Q. We find (using Convention 2.7)

a (a b) (a b c) = (a a b) (a a b c) = b (b c) = c,

a (a b) ((a b) (a c)) = a a c = c,

which implies a

(b

c) = (a

b)

(a

c) as ad

ab and ad

a are injective.We prove finally that is left distributive with respect to . Indeed, we have

a (a b) (a (b c)) = (a a b) (a a b c) = b (b c) = c,

a (a b) ((a b) (a c)) = a a c = c,

which implies a(bc) = (ab)(ac) as adab and ad

a are injective. A symmet-ric argument gives a(bc) = (ab)(ac).

Point (ii) is clear, as (2.3) follows from the definition of operation .

Owing to the previous results, we shall say that a left cancellative left divisibleLD-system is an implicit LD-quasigroup. There exists exactly one way tocomplete an implicit LD-quasigroup into an LD-quasigroup, so no ambiguity

exists.


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LD-quasigroups

Here we review examples of LD-quasigroups. In each case, we mention thoseresults about braids that can be deduced from the associated action.

Example 2.9. (trivial) Let S be an arbitrary set. Then the trivialoperations

a b = b, a b = b

turn S into an LD-quasigroup. Using such operations to colour thestrands of the braids amounts to keeping the colours unchanged. Then,for every braid b in Bn, and every sequencea in S

n, we have

a b = perm(b)1(a). (2.6)

The inversion comes from perm(b) specifying the initial positions of thestrands in terms of their final positions, i.e., going from bottom to top,as is needed forperm to be a homomorphism. Thus, using this particularLD-quasigroup leads to the surjective homomorphism

perm : Bn Sn.

Example 2.10. (shift) The operations

a b = b + 1, a b = b 1

turn Z into an LD-quasigroup. For every braid b in Bn, and every se-

quencea in Zn

, we find(a b) =

a + es(b), (2.7)

where

a denotesa1 + + an, and es(b) denotes the exponent sum of b,defined to be the difference between the number of positive and negativeletters in any expression of b. Thus, using this LD-quasigroup leads (in particular) to the homomorphism

es : Bn (Z, +).

This example as well as the previous one belong to the general type ab =f(b), ab = f1(b) with f a bijection of the considered domain.

Example 2.11. (mean) Assume that E is a Z[t, t1]-module. Thebinary operations

a b = (1 t)a + tb, a b = (1 t1)a + t1b (2.8)


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turn E into an LD-quasigroup. For every n-strand braid b, and everysequencea in En, the bottom coloursa b are linear combinations of the

top coloursa. There exists an n n-matrix rB(b) satisfyinga b = a rB(b), (2.9)

and using the current LD-quasigroup leads to a linear representation

rB : Bn GLn(Z[t, t1]). (2.10)

This linear representation is known as the (unreduced) Burau represen-tation. Example 2.9 (trivial) corresponds to the specialization t = 1.

Example 2.12. (conjugacy) Let G be a group. Then the binary oper-ations defined by

x y = xyx1, x y = x1yx (2.11)

turn G into an LD-quasigroup. In particular, let FGn be the free groupbased on {x1, . . . , xn}. For b an n-strand braid, let b

rev denote the imageof b under the antiautomorphism of B that is the identity on the gener-ators i: thus an expression of b

rev is obtained from an expression of b byreversing the orders of the letters. Then define elements y1, . . . , yn of Gby

(y1, . . . , yn) = (x1, . . . , xn) brev,

and let b be the endomorphism of FGn that maps xi to yi for every i.Then the mapping : b

b is an endomorphism of Bn into End(FGn),

and, as b1 = b1 holds by construction, the image of is actuallyincluded in Aut(FGn). As in Example 2.9 (trivial), using brev here isnecessary because we consider an action on the right, while, by definition,composition of automorphisms corresponds to an action on the left. Sousing the current LD-quasigroup leads to a representation

: Bn Aut(FGn).

It is known that this representation is faithful, i.e., is an embeddingthis will be proved in Chapter III.

Using free differential calculus, one can see the mean operations (2.8)as a projection of the operations (2.11), corresponding to the fact thatthe Burau matrix of a braid can be deduced from the associated auto-morphism of a free group.

The previous list more or less exhausts classical LD-quasigroups. A few moreare described in exercises below, but they are unessential variations. We see


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that using braid colourings and resorting to various (classical) LD-quasigroupsleads to several (classical) representation results for braid groups. It is therefore

natural to ask:

Question 2.13. Can we find new examples of LD-quasigroups, and deducefurther properties of braids?

The answer to the first part is positive, but it seems that the answer to thesecond part is essentially negative. To explain this, let us introduce one moreexample.

Example 2.14. (half-conjugacy) Let G be a group, and X be a subset

of G. The binary operations defined by

(a, x) (b, y) = (axa1b, y), (a, x) (b, y) = (ax1a1b, y) (2.12)

turn G X into an LD-quasigroup. This example is related to conjugacyas follows: let f be the mapping of G X to G defined by f((a, x)) =axa1; then f is a surjective homomorphism of (G X, , ) onto Gequipped with the conjugacy operations of Example 2.12.

The point is that the previous example is, in some sense, the most general one:indeed, we shall check in Chapter V that, if FGX is a free group based on X,then the LD-quasigroup consisting of FGX X with the operations (2.12) is a

free LD-quasigroup based on X. Hence every LD-quasigroup generated by Xis a quotient of FGX X, and, therefore, the most powerful results aboutbraids we can expect to obtain using LD-quasigroups are those coming fromFGX X. The mapping f of Example 2.14 is not an embedding, and, intheory, we could obtain more using FGX X with half-conjugacy than usingFGX with conjugacy, but the distance between the two structures is shorthalf-conjugacy is a sort of semidirect product of conjugacy with the operationsof Example 2.10 (shift)and no positive result in this direction is known.

On the other hand, the facts below suggest that LD-quasigroups are veryspecial examples of LD-systems, and, in particular, they suffer strong restric-tions in terms of orderability. Indeed, the LD-quasigroups of Examples 2.9(trivial), 2.11 (mean), and 2.12 (conjugacy) all are idempotent, i.e., they sat-

isfy the identityx x = x.

This is not true in the case of Examples 2.10 (shift) and 2.14 (half-conjugacy),but a weak form of idempotency still holds.


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Proof. Assume that (Q, ) is an LD-quasigroup. For every a in Q, we have(aa)a = aa, hence (aa) is a cycle of length 1 for left division.

Keeping our principle of using various LD-systems to colour the braids, andconsidering order features, we are led to the following double question:

Question 2.18. (i) Do there exist an LD-system (necessarily not an LD-quasigroup) where left division admits no cyclethus some sort of torsion-freeLD-system?

(ii) If so, can we use such an LD-system to colour braids?

We shall see in the sequel that the answer is positive in both cases. For themoment, let us conclude the section with one more example, which gives a(very) partial answer to Question 2.18(i).

Example 2.19. (injection bracket) Let X be a nonempty set. On theset IX of all injections of X into itself, we define a new operation by

f[g](n) =

f gf1(n) for n Im(f),n otherwise,

(2.14)

whereIm(f) denotes the image off. Below are displayed a few injectionsobtained in the case when X is N+ and the bracket is applied startingwith the shift mapping sh : n n + 1in the picture, the source is thebottom line, and the target is the top one.

sh

sh[sh]

sh[sh[sh]]

sh[sh][sh]

sh[sh][sh][sh].

Let us denote by coIm(f) the coimage of f, i.e., the complement of Imfin X, and by Gr(f) the graph of f, i.e., the set of all pairs (n, f(n)) forn X. The equality

Gr(f[g]) = f(Gr(g)) {(n, n); n coIm(f)} (2.15)


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which shows that (f, f[sh[4]]) is a cycle for left division in (I \ S, [ ]).

The previous example is our first non-classical LD-system. It has been in-spired by the LD-system associated with an elementary embedding describedin Chapter XII, and we shall connect it soon with other examples.

Exercise 2.21. (left and right equivalences) (i) Assume that (S, )is an LD-system. Write a

Lb for (x)(ax = bx). Prove that

L

is an equivalence relation on S that is compatible with on the right.Symmetrically, write a R b for (x)(xa = xb). Show that R is acongruence on (S, ), i.e., an equivalence relation on S that is compatiblewith on both sides.

(ii) Assume that (S, ) is a left divisible LD-system. Show that L

is

a congruence on (S,

).(iii) Assume that (Q, , ) is an LD-quasigroup. Write a

Lb for

(x)(ax = bx). Prove that L

is a congruence on (Q, , ), and thatthe quotient LD-quasigroup Q/

Lembeds in the LD-quasigroup made

of Aut(Q) equipped with the conjugacy operations (hence, in particular,it is idempotent). Deduce that every right cancellative LD-quasigroup isidempotent and it embeds in a group equipped with conjugacy. Extendthe result to every LD-quasigroup where

Lis trivial.

Exercise 2.22. (variations about conjugacy) (i) Assume that G is agroup. For i Z, define i by x i y = x

iyxi. Show that (G, i) is anLD-quasigroup, and i is left distributive with respect to j for all i, j.

(ii) Forf in End(G), definef on G byx f y = xf(yx1). Prove that(G, f) is an LD-system, and an LD-quasigroup if f is an automomor- phism. [A particular case isf being an inner automorphism of G: thisamounts to fixing an element a of G and defining x a y = xayx

1a1.](iii) Define on the set GZ of all Z-indexed sequences from G the oper-

ation so that xy = z holds for zi = xi yi+1 x1i+1. Show that (G

Z, ) isan LD-quasigroup.

(iv) For f in End(G), define on G by xy = xf(x1y). Show that(G, ) is an LD-system.

Exercise 2.23. (bilinear operations) Define on R3 by(x1, x2, x3)(y1, y2, y3) = (y1, y2, y3 + x1y2 x2y1).

Show that (R3, ) is an LD-quasigroup. [Hint: Observe that is thetranslation of conjugacy on unipotent 3 3-matrices.] Translate similarlythe operations of Exercise 2.22.


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(ii) Prove that (I, [ ]) is neither right cancellative, nor left or rightdivisible. Show that (I, [ ]) cannot be embedded in an LD-quasigroup.

(iii) WriteFix(g) for the set of all fixed points of g. Prove that (h)(g =f[h]) holds in I if and only if coIm(f) Fix(g) and g(Im(f)) Im(f)holds Deduce that g is left divisible by f if and only if f[g] = f[f][g]holds. [Hint: Assume f[g] = f[f][g]. Forn coIm(f), we have f[f](n) =f[g](n) = n, so f[g](n) = f[f][g](n) implies f[f](g(n)) = n, hence g(n) =n, and coIm(f) Fix(g). Assumen coIm(f). By the previous result,we have g(n) = n, hence assuming n g(Im(f)) implies n Im(f), acontradiction. Hence g(Im(f)) Im(f) holds.]

Exercise 2.29. (injection bracket) Consider the injection bracketon N+ N+. Let a and b respectively denote the first and the secondshift (p,q) (p + 1, q) and (p,q) q + 1. Establish a[b] = b[a] and

a[b[a[b]]] = a[a[b[b]]]. Prove that every element in the sub-LD-systemgenerated by a and b is a shift outside some finite rectangle.

Exercise 2.30. (generalized bracket) Let S denote the subset ofI2consisting of those pairs (f1, f2) satisfying f2(coIm(f1)) = coIm(f1).Show that the formula (f1, f2)[(g1, g2)] = (h1, h2), with

h1(n) = f1g1f11 (n), h2(n) = f1g2f

11 (n) for n Im(f1),

h1(n) = h2(n) = f2(n) otherwise,defines a left self-distributive operation on S.

Exercise 2.31. (Wadas identities) (i) Assume that and are binary

operations on S. We consider colourings obeying the rulea b

ab b a .

Show that the associated right action of BW+n on Sn is invariant under

positive braid relations if and only if(S, , ) satisfies the identitiesx(yz) = (xy)((y x)z),x (y z) = (x y) ((yx) z),((y x)z) (xy) = ((yz) x)(z y).(ii) Let G be a group. Show that the operations defined on G by

ab = a(ab)1, a b = (ab)b satisfy the above identities. Show that

completing the construction with ab = a(ab), a b = (ab)1b defines an

action of Bn on Gn. Generalize to ab = af(ab)1, ab = f(ab)b, where

f is an endomorphism of G satisfying f3 = f.


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3. A Self-Distributive Operation on Braids

In the previous section, we have seen how self-distributive systems can beused to obtain results about braids. In this section, we go the other way,and show how braids can be used to obtain results about self-distributivity byconstructing a left self-distributive operation on braids. The two approacheslook unrelated at first, but we shall see in Chapter VIII that both relie onone and the same fact, namely the connection between a certain group GLDassociated with self-distributivity and Artins braid group B.

Braid exponentiation

In the sequel, we consider braids with unboundedly many strands. For everyinteger n, the mapping that adds a trivial unbraided (n + 1)-th strand atthe right of a geometric n-strand braid induces an embedding in,n+1 of thegroup Bn into the group Bn+1. Then (Bn, in,n+1)n2 is a directed system.

Definition 3.1. (group B) The direct limit of the system (Bn, in,n+1) isdenoted B.

Then B is the group generated by an infinite sequence of generators 1, 2, . . .submitted to the braid relations (1.1) and (1.2); its elements can be representedby braid diagrams with strands indexed by the positive integers. Notice that,in every such diagram, only finitely many strands are braided.

Definition 3.2. (shift) We write sh for the shift endomorphism of B thatmaps i to i+1 for every i.

Lemma 3.3. The endomorphism sh of B is injective.

Proof. An algebraic argument is not obvious: if w, w are braid words andsh(w) sh(w) holds, we can deduce w w directly only if we know that allintermediate braid words in a sequence of elementary transformations from wto w belong to the image of sh. This is true if we restrict to positive words,but this need not be the case in general as new factors 1

11 or

11 1 may

appear. A geometric argument can be used here: for each braid diagram d,let us denote by coll(d) the diagram obtained from d by removing the strandthat begins at position 1. It is easy to check that d being isotopic to d impliescoll(d) being isotopic to coll(d). Hence, with obvious definitions, sh(d) sh(d)implies coll(sh(d)) coll(sh(d)), i.e., d d.


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Section I.3: A Self-Distributive Operation on Braids 27

Let us consider the question of constructing a left self-distributive operation

on B, called braid exponentiation in the sequel. In Chapter VIII, we shall

give deep arguments for such an operation to exist. For the moment, a heuristicapproach will lead us to a possible definition of. Let us assume that the latteroperation exists, and that we can use (B, ) to colour braidswhich a priorishould require the latter structure to be an LD-quasigroup. We introduce thenotion of a self-colouring braid as follows:

Definition 3.4. (self-colouring) (Assuming that a left self-distributive op-eration has been defined on B,) we say that a braid b is self-colouring if(1, 1, 1, . . .) b = (b, 1, 1, . . .) holds, i.e., if b produces itself starting from thecolours 1, 1, . . . (Figure 3.1)

1 1 1 1 1

b

b 1 1 1 1Figure 3.1. Self-colouring braid

As the existence and the definition of braid exponentiation remain open, whatself-colouring braids are is not clear. However, self-colouring braids do exist,for, by definition, at least the unit braid 1 is self-colouring. Now, let us try tofind a formula for operation so that self-colourings braids are closed under .The following result comes directly from Figure 3.2:

Lemma 3.5. Assume that a and b are self-colouring braids. Then we have

(1, 1, . . .) a sh(b) 1 sh(a1

) = (a

b, 1, 1, . . .).

1 1 1 1 1

a

a 1 1 1 1

b

a b 1 1 1

a1ab a 1 1 1

ab 1 1 1 1

Figure 3.2. Braid exponentiation

So the family of self-colouring braids will be closed under exponentiation pro-vided the latter is defined by the formula of Lemma 3.5. This leads us to:


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Definition 3.6. (braid exponentiation) For a, b in B, we define

a b = a sh(b) 1 sh(a1). (3.1)

Example 3.7. (braid exponentiation) The reader can check the fol-lowing equalities

1 1 = 1 = , 1 (1 1) = 21 = ,

(1 1) 1 = 2112 = .

From the above arguments, we can expect exponentiation to be self-distributiveon self-colouring braids. It being self-distributive on all of B is not obvious,but it is a matter of a simple computation.

Proposition 3.8. (LD-system) The set B equipped with exponentiationis a left cancellative LD-system.

Proof. We can either make a direct algebraic computation (cf. Exercise 3.19),or draw a picture (Figure 3.3) to check that a(bc) = (ab)(ac) holds forall a, b, c. Left cancellativity follows from sh being injective.

Although the definition of braid exponentiation resembles a skew conjugacy,

its properties are quite different from those of conjugacy, in particular as faras left division is concerned.

Proposition 3.9. (not LD-quasigroup) No equality ab = 1 holds in B.Hence, in particular, (B, ) is not an LD-quasigroup.

Proof. Let a, b be two braids with associated permutations f, g. Let p =f(1) + 1. Defining s1 to be the transposition that exchanges 1 and 2, we find

perm(a b)(p) = f sh(g) s1 sh(f)1(p)

= f sh(g) s1(2) = f sh(g)(1) = f(1) = p 1,

hence perm(a

b) is not the identity, and a

b cannot be the unit braid.

We refer to exercises for a complete study of division in (B, ).


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a

b

c

b1

a1

a

b

a1

a

c

a1

a

b1

a1

Figure 3.3. Self-distributivity of braid exponentiation

Derived operations

New left self-distributive operations can be deduced from braid exponentiation

by using representations of the braid group B. It need not be true that everygroup homomorphism of B induces a left self-distributive structure on theimage, but a weak compatibility condition is sufficient.

Lemma 3.10. Assume that r is a surjective group homomorphism definedon B. A sufficient condition for the exponentiation of B to induce a leftdistributive operation on the image of r is the inclusion Ker(r) Ker(r sh).

The verification is left to the reader. Let us consider some examples.

Example 3.11. (permutations) For every positive integern, the map-ping perm is a surjective homomorphism of Bn onto Sn. The symmetricgroups form a directed system when equipped with the embeddings in-duced by the identity mapping of {1, . . . , n} into {1, . . . , n + 1}, and thedirect limit is the group S of those permutations of N+ that move


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finitely many integers only. Then perm extends into a surjective homo-morphism ofB onto S, and the compatibility relation of Lemma3.10

is satisfied. So we obtain a left self-distributive operation on S bydefining

f g = f sh(g) s1 sh(g1), (3.2)

where s1 is the transposition (1 2) and sh is the shift endomorphismofS defined by sh(f)(1) = 1 and sh(f)(n) = f(n 1) + 1 for n > 1.

Example 3.12. (Burau matrices) The Burau representation of Bninto GLn(Z[t, t1]) has been introduced in Example2.11 (mean). It easilyextends into a representation of B. Let in,n+1 and shn denote theembeddings of GLn(Z[t, t

1]) into GLn+1(Z[t, t1]) defined respectively

by

in,n+1 : M

0

M...0

0 0 1

, shn : M

1 0 00... M0

.The direct limit GL(Z[t, t

1]) of the system (GLn(Z[t, t1]), in,n+1) isthe union of the groups GLn(Z[t, t

1]) turned into a group: the productof two matrices with different sizes is obtained by first completing thesmaller matrix trivially on the right and the bottom. Then the map-pings shn induce a well-defined endomorphism of GL(Z[t, t

1]). TheBurau representation rB of B is defined by the rules

rB(1) = 1 t t1 0 , rB(1+i) = shi((1)).The condition of Lemma 3.10 is fulfilled, so carrying the exponentiationof B gives a distributive operation on the image of rB. The latter set isa proper subset of GL(Z[t, t

1]), but we can consider the formula thatdefines the operation, namely

A B = A sh(B) rB(1) sh(A1), (3.3)

for A, B ranging on all of GL(Z[t, t1]), and it is easy to check left

self-distributivity directly.So we have found a left distributive operation on matrices. Starting

from the unit, which, under our conventions, is the 1, 1-matrix (1), we

can compute simple expressions like

(1) (1) =

1 t t

1 0

, (1) ((1) (1)) =

1 t t 01 t 0 t1 0 0

,


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((1) (1)) (1) =

1 t + t2 0 t t2

1 t 0 t

0 t

1

1 t1

.

Specializing t (with t = 0) gives a left distributive operation on matri-ces with complex, real or integer entries. In particular, t = 1 gives theexponentiation of permutations described above.

Example 3.13. (automorphisms of free groups) We have seen inSection 2 that letting the braid group Bn act on a free group FGn of rankngives a homomorphism ofBn into the group Aut(FGn). Again, the con-struction is compatible with the canonical embedding of Bn into Bn+1, so,ifFG denotes the free group based on the infinite sequence {x1, x2, . . .},we obtain a homomorphism, still denoted , of B into Aut(FG). Byconstruction, the automorphism i is determined by

i(xk) = xk for k < i or k > i + 1,xixi+1x1i for k = i,xi for k = i + 1.

(3.4)

We write sh for the shift endomorphism of FG that maps xi to xi+1for every i, and also for the shift endomorphism of the group Aut(FG)defined as follows: for f in Aut(FG), sh(f) is the automorphism de-fined by sh(f)(x1) = x1 and sh(f)(xi+1) = sh(f(xi)) for i 1. Forinstance, we have sh(i) = i+1 for every i. Copying the definition ofbraid exponentiation leads to introducing for f, g in Aut(FG)

f g = f sh(g)

1 sh(f

1). (3.5)

It is easy to verify that this operation is left self-distributive on Aut(FG),and that is a homomorphism of (B, ) into (Aut(FG), ).

Cycles for left division

As mentioned above, one of the restrictions suffered by all LD-quasigroups isa form of torsion, namely that left division admits lots of cycles, even cyclesof length 1. However, we have seen that the bracket operation on injectionsyields an LD-system where left division admits no cycle of length 1. In thissection, we prove a much stronger statement in this direction, namely that leftdivision for the exponentations of Aut(FG) and of B admit no cycle. SuchLD-systems are therefore quite different from idempotent LD-systems, and,more generally, from all previously known examples. This fact will b e crucialfor the construction of a braid order in Chapter III.


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Proposition 3.14. (automorphisms acyclic) Left division in Aut(FG)equipped with the operation of (3.5) admits no cycle.

The result follows from two lemmas about possible free reductions occurringwhen computing the image of x1. In the sequel, we write X for {x1, x2, . . .},and X1 for {x11 , x

12 , . . .}. For w a word on X X

1, we say that w is freelyreduced if it contains no pattern xix

1i or x

1i xi, and, in every case, we denote

by red(w) the unique reduced word that can be obtained from w by iterativelydeleting the patterns xix

1i or x

1i xi. Then the free group FG identifies with

the set of all freely reduced words equipped with the operation u v = red(uv).

Definition 3.15. (sets S(x)) For x in X X1, we define S(x) to be thesubset of FG consisting of all reduced words that finish with the letter x.

Let us investigate the image of the set S(x11 ) under the automorphism

1i .

Lemma 3.16. For every f in Aut(FG), sh(f) maps S(x11 ) into itself.

Proof. Let us consider an arbitrary element of S(x11 ), say wx11 , with w /

S(xi). By construction, we have sh(f)(wx11 ) = red(sh(f)(w) x

11 ). Assume

sh(f)(wx11 ) / S(x11 ). Then the final letter x

11 in sh(f)(w)x

11 is cancelled

by some letter x1 occurring in sh(f)(w). Such a letter x1 in sh(f)(w) mustcome from a letter x1 in w. So there exists a decomposition w = w1x1w2satisfying sh(f)(w2) = 1. As sh(f) is injective, the latter condition impliesw2 = 1, hence w S(x1), contradicting the hypothesis.

Lemma 3.17. The automorphism 1 maps S(x11 ) into itself.Proof. Let us consider again an arbitrary element of S(x11 ), say wx

11 , with

w / S(xi). We have 1(wx11 ) = red(1(w) x1x12 x11 ). Assume 1(wx11 ) /S(x11 ). Then the final x

11 in 1(w) x1x12 x11 is cancelled by some letter x1

in 1(w), which comes either from some x2 or from some x11 in w.In the first case, we display the letter x2 involved and write w = w1x2w2,

with w2 a reduced word not beginning with x12 . We find1(wx11 ) = red(1(w1) x1 1(w2) x1x12 x11 ),

and the hypothesis red(1(w2)x1x12 ) = implies 1(w2) = x2x11 =1(x12 x1). As 1 is injective, we deduce w2 = x12 x1, contradicting the hy-pothesis that w2 does not begin with x

12 .

In the second case, we write similarly w = w1xe1w2 with e = 1. We find

now

1(wx11 ) = red(1(w1) x1xe2x11 1(w2) x1x12 x11 ),and the hypothesis red(xe2x

11 1(w2)x1xe2) = implies 1(w2) = x1x1e2 x11 =1(x1e1 ), hence w2 = x1e1 , and either w2 = or w2 = x21. In both cases, w

belongs to S(x1), a contradiction.


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Lemma 3.18. Assume that f is an element ofAut(FG) that can be expressedas a product of elements in the image of sh and of

1, the latter occurring at

least once. Then we have f(x1) S(x1

1 ), and, therefore, f = id.

Proof. The hypothesis is that f admits a decomposition of the form

f = sh(f0) 1 sh(f1) 1 sh(fp).Then we have sh(fp)(x1) = x1, and 1(x1) = x1x2x11 , an element of S(x11 ).By Lemmas 3.16 and 3.17, every subsequent factor sh(fk) and 1 maps S(x11 )into S(x11 ).

We can now prove Proposition 3.14 (automorphisms acyclic). We have to showthat no equality of the form

f = ( (f g1) ) gp (3.6)

is possible in Aut(FG). By using the explicit definition of the operation

on Aut(FG), we expand (3.6) into an equality of the form

f = f sh(f0) 1 sh(f1) sh(fp1) 1 sh(fp),which implies id = sh(f0)1 sh(f1) sh(fp1)1 sh(fp). By Lemma 3.18,such an identity is impossible.

Proposition 3.19. (braids acyclic) (i) Left division in B equipped withbraid exponentiation admits no cycle.

(ii) Assume that the braid b admits an expression where1 occurs but 11

does not. Then b = 1 holds.

Proof. (i) We know that there exists a homomorphism of (B, )into (Aut(FG), ). The image under of a possible cycle in the divisionof (B, ) would be a cycle in the division of (Aut(FG), ), contradictingProposition 3.14 (automorphisms acyclic).

Point (ii) follows from Lemma 3.18 directly. If b is a braid admitting an

expression where 1 occurs but 11 does not, the associated automorphism

bis eligible for Lemma 3.18, so we have b = id, and, therefore, b = 1. Observe that the previous arguments do not require to be injective (which willbe proved in Chapter III). On the other hand, they do not transfer to quotientsof B such as S, since quotienting may add cycles in the divisionand itdoes in the case ofS, as shown in Exercise 3.27 below.

Exercise 3.20. (braids are unavoidable) Assume that G is a group,a is a fixed element of G, and s is an endomorphism ofG. Prove that the


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Section I.4: Extended Braids 35

injective on B.

Exercise 3.26. (Burau exponentiation) For M a matrix, say M =(mi,j), define Tr

+(M) to be

i mi,i+1. Prove Tr+(AB) = Tr+(B) + t.

Exercise 3.27. (exponentiation of permutations) (i) Show that leftdivision in (S, ) admits no cycle of length 1. [Hint: f = fg wouldimply s1 = sh(g

1 f).]

(ii) Check the equalitiesidid = s1, (idid)id = s2, id(idid) = s2s1,((idid)id)id = s2s1s3, where si is the transposition (i, i + 1).

(iii) Let f = s1s3 and g = s2s1. Show that (f, g) is a cycle of length 2for left division in (S, ) [Hint: prove f = gs1 and g = f(s2s1s3)].Show that f and g belong to the sub-LD-system of (S, ) generatedby id. [Hint: We have f = s2s1.]

(iv) Prove that the mappingf fsh defines an embedding of(S, )into (I, [ ]). Describe the image.

4. Extended Braids

Several extensions of the braid groups appear naturally when the approachof the previous sections is developed. Here we introduce one such extensionstarting from the idea of colouring braid diagrams in such a way that both thestrands and the regions between the strands receive a colour.

Region colourings

Let (S, ) be an LD-system. For every (positive) braid

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