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Artin groups and local indicability Mulholland, Jamie Thomas 2002

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ARTIN GROUPS AND LOCAL INDICABILITY by JAMIE THOMAS M U L H O L L A N D B.Sc. Simon Fraser University, 2000 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept#fts-thesis as conforming to th\e require^ standard THE UNIVERSITY OF BRITISH COLUMBIA September 2002 © Jamie Thomas Mulholland, 2002 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for exten-sive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date October 7, SOOSL Abstract This thesis consists of two parts. The first part (chapters 1 and 2) consists of an introduction to theory of Coxeter groups and Artin groups. This material, for the most part, has been known for over thirty years, however, we do mention some recent developments where appropriate. In the second part (chapters 3-5) we present some new results concerning Artin groups of finite-type. In particular, we compute presentations for the commutator subgroups of the irreducible finite-type Artin groups, generalizing the work of Gorin and Lin [GL69] on the braid groups. Using these presentations we determine the local indicability of the irreducible finite-type Artin groups (except for F4 which at this time remains undetermined). We end with a discussion of the current state of the right-orderability of the finite-type Artin groups. i i Table of Contents Abstract , i i Table of Contents i i i List of Tables vi List of Figures vi i Acknowledgement vi i i Dedication ix Chapter 0. Introduction and Statement of Results 1 0.1 Introduction : 1 0.2 Outline and Statement of Results 2 Chapter 1. Basic Theory of Coxeter Groups 4 1.1 Definition ' 4 1.2 Length Function 6 1.3 Geometric Representation of W 7 1.4 Root System 8 1.5 Strong Exchange Condition 10 1.6 Parabolic Subgroups 11 1.7 The Word and Conjugacy Problem 14 1.8 Finite Coxeter Groups • • 15 Chapter 2. Basic Theory of Artin Groups 18 2.1 Definition 18 2.2 Positive Artin Monoid 19 2.3 Reduction Property 20 2.4 Divisibility Theory 22 2.4.1 Chains 22 2.4.2 Chain Operators Ka :'. 24 2.4.3 Division Algorithm 27 2.4.4 Common Multiples and Divisors 28 iii Table of Contents iv 2.4.5 Square-Free Positive Words : — 32 2.5 The Fundamental Element 34 2.6 The Word and Conjugacy Problem 35 2.7 Parabolic Subgroups 36 2.8 Geometric Realization of Artin Groups 38 Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 40 3.1 Reidemeister-Schreier Method . . . 40 3.2 A Characterization of the Commutator Subgroups 41 3.3 Computing the Presentations : 44 . 3.3.1 Two Lemmas 44 3.3.2 Type A 50 3.3.3 TypeB 54 3.3.4 TypeL> 60 3.3.5 Type£ 61 3.3.6 TypeF . . . .62 3.3.7 Type H :. 63 3.3.8 Type/ 64 3.3.9 Summary of Results , 64 Chapter 4. Local Indicability of Finite-Type Artin Groups 66 4.1 Definitions 66 4.2 The Local Indicability of Finite-Type Artin Groups 68 4.2.1 Type A 69 4.2.2 TypeS , . ' 69 4.2.3 Type/J ..70 4.2.4 Type£ .70 4.2.5 TypeF 70 4.2.6 TypeF 71 4.2.7 Type / 71 Chapter 5. Open Questions: Orderability 72 5.1 Orderable Groups 72 5.2 Finite-Type Artin Groups — 74 5.2.1 Ordering the Monoid is Sufficient 74 5.2.2 Reduction to Type E8 76 iv Table of Contents i Bibliography 77 Index 82 • r ( List of Tables 1.1 Inclusions among Coxeter groups 17 2.1 Inclusions among Artin groups 37 3.1 Properties of the commutator subgroups 65 vi f List of Figures 1.1 A l l the connected positive definite Coxeter graphs 16 3.1 Focid for the irreducible finite-type Coxeter graphs V 45 vii Acknowledgement ^ It gives me great pleasure to thank the many people and organizations who have helped me to get to where l a m today. I am very grateful for financial support from the University of British Columbia and from NSERC. To the many teachers who have guided me to where I am today:— thanks for your knowledge, wisdom, and inspiration. I owe an enormous debt to my supervisor, Dr. Dale Rolfsen, for having guided me through my MSc, sharing his knowledge, wisdom and experience of mathematices with me on the way. My wonderful girlfriend, Heather Mosher, who has offered love and sup-port through thick and thin - my deepest thanks. And my family, whose love and support, encouragement and guidance, have always been complete, and whose belief in me has enabled me to get to this point, the warmest thanks of all. viii i To my Mother and Father ix Chapter 0 Introduction and Statement of Results 0.1 Introduction A number of recent discoveries regarding the Artin braid groups Q3n com-plete a rather interesting story about the orderability1 of these groups. These discoveries were as follows. In 1969, Gorin and Lin [GL69], by computing presentations for the com-mutator subgroups *8'n of the braid groups Q3n, showed that 233 is a free group of rank 2, is the semidirect product of two free groups (each of rank 2), and 93^  is finitely generated and perfect for n > 5. It follows from these results that <Bn is locally indicable2 if and only if n = 2,3, and 4. Neuwirth in 1974 [Neu74], observed <8n is not bi-orderable if n > 3. How-ever, Patrick Dehornoy [Deh94] showed the braid groups are in fact right-orderable for all n. Furthermore, Dale Rolfsen and Jun Zhu [RZ98] proved (non-constructively3) that the subgroups Vn of pure braids are bi-orderable. So, by this point in time (1998), the orderability of the braid and pure braid groups were known. What remained unknown was the relationship between a right-ordering on !B n and a bi-ordering on Vn- That is, does a right-ordering on restrict to a bi-ordering on Vn? This question was recently answered by Rolfsen and Rhemtulla [RR02] J A group G is right-orderable if there exists a strict total ordering < of its elements which is right-invariant: g < h implies gk < hk for all g,h,k e G. If in addition g < h implies kg < kh, the group is said to be orderable, or for emphasis, bi-orderable. 2A group G is locally indicable if for every nontrivial, finitely generated subgroup there exists a nontrivial homomorphism into Z (called an indexing function). 3Rolfsen and Djun Kim construct a bi-ordering on Vn in [KR02]. 1 Chapter 0. Introduction and Statement of Results 2 by determining the connection between local indicability and orderability. In particular, they showed that since the braid groups *Bn are not locally indi-cable for n > 5 a right-ordering on 25n could not restrict to a bi-ordering on •p 4 ' This thesis is concerned with investigating whether these results on the braid groups extend to all finite-type Artin groups. In particular, we are con-cerned with determining the local indicability of the finite-type Artin groups. 0.2 Outline and Statement of Results In Chapter 1 we give a quick yet thorough introduction to the theory of Cox-eter groups. In Chapter 2 we introduce Artin groups and develop their basic theory. Most of these results have been known for over thirty years, however, we do mention recent developments where appropriate. The remaining chapters consist of recent and new results. In Chapter 3 we follow the direction of Gorin and Lin and compute pre-sentations of the commutator subgroups of the finite-type Artin groups. The results here are new (aside from the particular case of the braid groups which were done, of course, by Gorin and Lin). In Chapter 4 we use these presentations to extend the results of Gorin and Lin on the braid groups to the class of finite-type Artin groups as follows. Theorem 0.1 The following are finitely generated and perfect: !• A'Anforn>4, 2. A'Bnfor n > 5, ~ • 3. A'DJorn>t>, 4. A'Enforn = 6,7,8, 5. A'Hnforn = 3,4. Hence, the corresponding Artin groups are not locally indicable. 4see theorem 5.7. Chapter 0. Introduction and Statement of Results 3 On the other hand, we show the remaining finite-type Artin groups are locally indicable (excluding the type F 4 which at this time remains undeter-mined) . In Chapter 5 we discuss the orderability of the finite-type Artin groups. We show that in order to determine the right-orderability (bi-orderability) of the finite-type Artin groups it is sufficient to determine whether the positive Artin monoid is right-orderable (bi-orderable). Furthermore, we show that in order to prove all finite-type Artin groups are right-orderable it suffices to show the Artin group of type Eg is right-orderable. Chapter 1 Basic Theory of Coxeter Groups The first comprehensive treatment of finite reflection groups was given by H . S.M. Coxeter in 1934. In [Cox34] he completely classified the groups and derived several of their properties, using mainly geometrical methods. He later included a discussion of the groups in his book Regular Poly topes [Cox63]. Another discussion, somewhat more algebraic in nature, was given by E. Witt in 1941 [Wit41]. A more general class of groups; the Coxeter groups, to which finite reflection groups belong, has since been studied in N . Bourbaki's chap-ters on Lie Groups and Lie Algebras [Bou72], [Bou02]. Another discussion appears in Humphrey's book Reflection Groups and Coxeter Groups [Hum72]. In this chapter we develop the theory of Coxeter groups with emphasis on the "root system" (following Deodhar [Deo82]). The approach we take here is precisely that of Humphreys [Hum72].. A l l of the results found in this chapter may be found in some form or another in Humphreys book , however, its inclusion here has primarily two purposes: (1) to make this thesis self contained for the convience of the reader and (2) to draw a comparison with the theory of Artin groups developed in chapter 2. The material has been reorganized and emphasis has been put on the parts of the theory we wish to compare with the theory of Artin groups. I. 1 Definition Let S be a finite set. A Coxeter matrix over 5 is a matrix M = (mss>)SjSies indexed by the elements of S and satisfying • (a) mss = 1 if s € S, 4 Chapter 1. Basic Theory of Coxeter Groups 5 (b) mss> = msis E {2, . . . , oo} if s,s' E S and s ^ s'. A Coxeter matrix M = (mss')S:S>es is usually represented by its Coxeter graph F. This is defined by the following data. (a) S is the set of vertices of F. (b) Two vertices s,s' E S are joined by an edge if mssi > 3. (c) The edge joining two vertices s,s' E S is labelled by mss> if mss> > 4. The Coxeter system of type F (or M) is the pair (W, S) where W is the group having the presentation W = (sES : (ss')m°°' = 1 if mssl < oo). The cardinality |5| of 5. is called the rank of (W, S). The canonical image of S in W is a generating set which may conceivably be smaller than S, that is, under the above relations two generators in S may be equal in W. In 1.3 we show this does not happen. Furthermore, we show in theorem 1.14 that no proper subset of 5 generates W. In the meantime, we may allow ourselves to write s E W for the image of s E S, whenever this creates no real ambiguity in the arguments. We refer to W itself as a Coxeter group of type T (or M), . when the presentation is understood, and denote it by Wp. Although a good part of the theory goes through for arbitrary S, we shall always assume that S is finite. However, this does not mean that the Coxeter group W is finite. Here are a couple of examples. Example 1.1 Ifmssi = oo when s ^ s' then W is the free product of\S\ copies of ,Z/2Z. This group is sometimes referred to as a universal Coxeter group. Example 1.2 It is well known that the symmetric group on (n + l)-letters is the Coxeter group associated with the Coxeter graph; • • • • • • > • • 1 - 2 • 3 n - 2 n - 1 n where vertex i corresponds to the transposition (i i + 1). When a group is given in terms of generators and relations it is quite dif-ficult to say anything about the group - for example, is the group trivial or Chapter 1. Basic Theory of Coxeter Groups 6 not? In our case it is quite easy to see that W has order at least 2. Con-sider,the map from S into {±1} , defined by taking each element of S to —1. Since this map takes each relation (ss')m^' to 1 it determines a homomor-phism e : W —> {±1} sending the image of each s G S to — 1. The map e is the generalization for an arbitrary Coxeter group of the sign character of the symmetric group. Theorem 1.3 There is a unique epimorphism e : W —> {±1} sending each gener-ator s G S to —1. In particular, each s has order 2 in W. Note that when \S\.= 1, W is just a group of order 2, i.e. Z/2Z. When \S\ = 2, say S = {s, s'}, W is the dihedral group of order 2mssi < oo. 1.2 Length Function We saw that the generators s G S have order two in W, so each w ^ 1 in W can be written as a word in the generators with no negative exponents: w = s\S2 • • • sr for some Sj (not necessarily distinct) in S. If r is as small as possible we call it the length of w, written l(w), and we call any expression of w as a product of r elements of S a reduced expression. By convention 1(1) = 0. Note that if s\S2---sr is a reduced expression then so are all initial segments, i.e. S1S2 • • • Si, i < r. Some basic properties of the length function are included in the following lemma, whose proof is straightforward. Lemma 1.4 The length function I has the following properties: (LI) l(w) = l(w~l), . (12) l(w) = lijfw£S, (L3) l(ww') < l(w) + l(w'), (LA) l(ww') > l(w) - l(w'), (L5) l(w) - V< l(ws) < l(w) + l,forseS and w G W. Property (Lb) tells us that the difference in the lengths of ws and w is at most 1, the following theorem tells us that this difference is exactly 1. Theorem 1.5. The homomorphism e : W — • {±1} of theorem 1.3 is given by e(w) = (-l)l(-w\ Thus, l(ws) = l(w) ± 1, for alls G S and w G W. Similarly for l(sw). Chapter 1. Basic Theory of Coxeter Groups 7 Proof. Let w € W have reduced expression si«2 • • • sr, then e(w) = e(Sl)e(s2)---e(sry=(-iy = (-iyW. Now e(ws) = e(w)e(s) = —e(w) implies l(ws) ^ l(w). • In our study of Coxeter groups we will often use induction on l(w) to prove theorems. It will therefore be essential to understand the precise re-lationship between l(w) and l(ws) (or l(sw)). It is clear that if w <G W has a reduced expression ending i n s e 5 then l(ws) = l(w) — 1, however it is not clear at this point whether the converse is true: for w € W and s € S if l(ws) = l(w) — 1 then w has a reduced expression ending'in s. This turns out to be true, see section 1.5, but to prove this we need a way to represent W concretely. 1.3 Geometric Representation of W Since Coxeter groups are generalizations of finite orthogonal reflection groups it should be no surprise that we wish to view W as a "reflection group" on some real vector-space V. It is too much to expect a faithful repre-sentation of W as a group generated by (orthogonal) reflections in a euclidean space. However, we can get a reasonable substitute if we redefine a reflection to be merely a linear transformation which fixes a hyperplane pointwise and sends some nonzero vector to its negative. Define V to be the real vector space with basis {as : s € S} in one-to-one correspondence with S. We impose a geometry on V in such a way that the "angle" between as and ay will be compatible with the given m s s / . To do this, we define a symmetric bilinear form B on V by requiring B(as, a , = — cos . mss> In the case of mssi — oo the expression is interpreted to be —1. From this definition we have B(as,as) = 1, while B(as,a's) < 0 for s ^ s'. Note that B is not necessarily -positive definite, i.e. there are Coxeter groups W for which some v € V does not satisfy B(v, v) > 0. Consider the following example. Chapter 1. Basic Theory of Coxeter Groups 8 Example 1.6 For the universal Coxeter group of rank two> W = ( s i , s 2 : si,si), take v = aSl + aS2 G V. It is easy to check B(aSl + aS2, aSl + aS2) = 0. Moreover, the following example shows that B may not even be positive semidefinite. Example 1.7 For the Coxeter group W = {si, s2, s 3 : si, s2,, si, (sis2)4, ( s i s 3 ) 4 , (s 2 s 3 ) 4 ) , take v — asi + aS2 + aS3 G V. Since B(aSi, aSj) = — cos f < -1 for i ^ j, then B(v,v) < - 1 . i For each s G S we can now define a reflecton as : V —> V by the rule: as(A) = \-2B(as,\)as. Clearly as(as) = —as, while as fixes i f s = {A G V : B(as, A) = 0} pointwise. In particular, we see that as has order 2 in GL(V). Theorem 1.8 There is a unique homomorphism a : W —• GL(V) sending s to as, and the group a(W) preserves the form B on V. Moreover, for each pair s, s' G 5, the order of ss' in W is precisely mss> For a proof of this theorem see Humphreys [Hum72]. To avoid cumber-some notation, we usually write w(as) to denote a(w)(as). The last statement in the theorem removes the possibility of s = s' in W even though s ^ s' in S, as promised in section 1.1. We will show next that this representation is indeed a faithful one. To do this we need to introduce the concept of a root system. 1.4 Root System For a Coxeter system (W, S) a root system $ of W is a set of vectors in V satisfying the conditions: (RI) $ n R a = { ± a } for all a G $ (R2) s$ = $ for all s G S Chapter 1. Basic Theory of Coxeter Groups 9 The elements of $ are called roots. We will only be concerned with the specific root system given by $ = {w(cts) : w G W, s G S}. It is clear that axiom (R2) is satisfied for this choice of $, to check axiom (RI) it suffices to note that since W (more precisely o(W)) preserves the form B onV (theorem 1.8), $ is a set of unit vectors. Note that $ = —<& since if /3 = w(as) G <fr then — f3 = ws(as) is also in 3>. If a is any root then it can be expressed in the form OJ = E C s a s ( C s € R). ses If cs > 0 for all s G S then we call a a positive root and write a > 0. Similarly, if cB < 0 for all s € S then we call a a negative root and write a < 0. We write $ + and $~ for the respective sets of positive and negative roots. It may come as some surprise that these two sets exhaust this follows from the following theorem. The proof of this theorem is nontrivial, we refer the reader to Humphreys [Hum72] for proof. The set of roots {as : s € S} are called simple roots. Theorem 1.9 Let w e W and s G S. Then l(ws) > l(w) iff w(as) > 0. Equivalently, l(ws) < l(w) iff w(as) < 0. This tells us the precise criterion for l(ws) to be greater than l(w): w must take as to a positive root. This is the key to all further combinatorial proper-ties of W relative to the generating set 5. Corollary 1.10 The representation a : W —> GL(V) is faithful. Proof. Let w G Ker(a). If w ^ 1 then it has reduced expression s\S2 • • • sr where r > 1. Since l(wsr) = r - 1 < l(w) then w(aSr) < 0 by theorem 1.9. But W(O>ST) = o S r > 0, which is a contradiction. • Another consequence of Theorem 1.9 is that the length of w e W is com-pletely determined by how it permutes For w G W let IT(w;) denote the set of positive roots sent to negative roots by w, i.e U(w) — {a G $ + : w(a) < 0}. Chapter 1. Basic Theory of Coxeter Groups 10 Theorem 1.11 (a) If s G S, then s sends as to its negative, hut permutes the re-maining positive roots. That is,U(s) = {as}. (b)Forallw eW,l{w) = \U(w)\. This theorem provides valuable information about the internal structure of W, see section 1.5. We refer the reader to Humphreys [Hum72] for the straightforward proof. If W is infinite the length function takes on arbitrarily large values (recall we are assuming S is finite). It follows from theorem 1.11 that $ is infinite. One the other hand, if W is finite ($ is also finite by definition) it contains a unique element of maximal length . Indeed, clearly W must contain at least one element of maximal length, say wo- For s G S, l(wos).< l(wo) so wo(as) < 0. Thus, WQ sends all positive roots to negative roots, i.e. U(WQ) = $ + . Sup-pose that there is another element w\ G W of maximal length, then u>f1 is also of maximal length and so n(u;^ 1) = It follows that wow^1(^+) = $ + , so / ( w o ^ r 1 ) = 0- Therefore wo — w\ so we have uniqueness. Since WQ and WQ1 have the same length uniqueness of the maximal element implies wo = WQ1, moreover it follows from theorem 1.11 that l(wo) = |$+|. 1.5 Strong Exchange Condition We are now in a position to prove some key facts about reduced expressions in W, which is at the heart of what it means to be a Coxeter group. Theorem 1.12 (Exchange Condition) Let w = sx • • • sr (si G S), not necessarily a reduced expression. Suppose a reflection s G S satisfies l(ws) < l(w). Then there is an index ifor which ws = s\ - ••§}••• sr (omiting s j . If the expression for w is reduced, then i is. unique. There is a stronger version of this theorem, called the Strong Exchange Condition in which the simple reflection s can be replaced by any element w G W which acts on V as a reflection, in the sense that there exists a unit vector a G V for which w(\) = A - 2B(X, a)a. It turns out that the vector a must be a root for w to act on V in this way. On the other hand, to each positive root a G $ + there is a w G W which acts on V as a reflection along a. Indeed, take w' G W, s G S such that a — w'(as). Then w = w's(w')~1 is Chapter 1. Basic Theory of Coxeter Groups 11 such an element'. Thus, there is a one-to-one correspondence between the set of positive roots $ + and the set of reflections in W. For a complete discussion see Humphreys ([Hum72] sec. 5.7,5.8). Before we prove theorem 1.12 we need to make the following observa-tion. If s, s' G S and w G W satisfy ay = w(as) then wsw~l —. s'. Indeed, tusu;~1(A) = w(w~1(X) - 2B(w'1 (X), as)as) and since B is W-invariant the result follows. Proof. Since l(ws) < l(w) then w(as)' < 0. Because a s > 0 there exists an index i < r for which SJ + I • • • sr(as) > 0 but SjSj+i • • • sr(as) < 0. From theorem 1.11 we have Sj+i • • • sr(as) = aSi, and by the above observation Si+i • • • s r s s r • • • Si+i = Si, from which it follows ws = si • -.- s~i • • • sr. In case l(w) = r consider what would happen if there were two distinct indices i < j such that ws = si • • • si • • • sr = si • • • s~j • • • sr. After cancelling, this gives s , + i • • • Sj = S f • Sj-i, or Si • • • Sj = s^+i • • • Sj-±, allowing us to write w = si • • • si • • • s~j • •'• sr. This contradicts l(w) = r. • Corollary 1.13 (a) (Deletion Condition,) Suppose w = si • • • sr (si G S), with l(w) < r. Then there exists i < j such that w = s\ • • • s} ••• • Sj • • • sr. (b) If w = s\ • • • sr, (si G S), then a reduced expression for w may be obtained by omitting on even number of S j . Proof. (a) There exists an index j such that l(w'sf) < l(w') where w' = s\---Sj-i. Applying the exchange condition gives W'SJ = s i • • • si • • • Sj-i, allowing us to write w = W'SJ • • • sr = s i • • • si • • • s~j • • • sr. • 1.6 Parabolic Subgroups In this section we show that for a Coxeter system (W, S) the subgroup of W generated by a subset of S is itself a Coxeter system with the obvious Coxeter graph. Let (W, S) be a Coxeter system with values mssi for s, s' G S. For a subset I C S we define Wj to.be the subgroup of W generated by / : At the extremes, = 1 and Ws = W. We call the subgroup Wj a parabolic subgroup . (More generally, we refer to any conjugate of such a subgroup as a parabolic Chapter 1. Basic Theory of Coxeter Groups 12 subgroup.) Let li denote the length function on Wj hvterms of the generators I. Theorem 1.14 (a) Tor each subset Iof S, the pair (Wi, I) is a Coxeter system with the given values mss'. (b) Let Id S. If w = s\- • • sr (si G S) is a reduced expression, and w G Wi, then all Si G / . In particular, the function I agrees with the length function li on Wj, and Wins = i. (c) The assignment 1i—• Wi.defines a lattice isomorphism between the collection of subsets of S and the collection of subgroups Wi ofW. (d) S is a minimal generating set for W. Proof. For (a). The set I and the corresponding values mss> give rise to an abstractly defined Coxeter group Wj, to which our previous results apply. In particular, Wi has a geometric representation of its own. This can obviously be identified with the action of the group generated by all as (s G /) on the subspace Vi of V spanned by all as (s G /), since the bilinear form B restricted to Vi agrees with the form Bi defined by Wj. The group generated by these as is just the restriction to Vj of the group cr(Wi). On the other hand, Wi maps canonically onto Wi, yielding a commutative triangle: Wi — • GL(V) \ / Wi Since the map Wi —> GL(Vi) is injective by corollary 1.10, we conclude that Wi is isomorphic to Wi and is therefore itself a Coxeter group. For (b), use induction on l(w), noting that Z(l) = 0 = Z/(l). Suppose w ^ 1 and let s = sr. Since w G Wi it also has a reduced expression w = t\ • • • tq, where ti G I. Now, i w(as) = as + ^ 2ciati (CJ G R) . i=i • ' According to theorem 1;9 l(ws) < l(w) implies w(as) < 0, so we must have ti = s for some i, forcing s E l . Now, ws = s\ • • • s r _ i G Wi, and the-expres-sion is reduced. The result follows by induction. Chapter 1. Basic Theory of Coxeter Groups 13 To prove (c), suppose I, J c S. If W> C Wj, then, by (b), J = W> n 5 C Wj n 5 = J Thus 7 c J (resp. / = J) if and only if W 7 c Wj (resp. Wj — Wj). It is clear that W / U J is the subgroup generated by W> and Wj. On the other hand, (b) implies that W / n j = Wj n Wj. This yields the desired, lattice isomorphism. To prove (d), suppose that a subset / of S generates W then Wj = W = Ws, so by (c) I = S. • If T is the Coxeter graph associated with the Coxeter system (W, S) then theorem 1.14 tells us that the Coxeter graph associated with (Wj,I) is pre-cisely Tj: the subgraph induced by I, that is, the subgraph of T with vertex set I and all edges (from F) whose endpoints are in / . Another way to view this result is that every induced subgraph of F is a Coxeter graph for some (parabolic) subgroup of W. We say that the Coxeter system (W, S) is irreducible if the Coxeter graph is connected. In general, let Fx, ..., Fr be the connected components of F, and let ij be the corresponding sets of generators from S, i.e. the verticies of Tj. Thus if s G Ii and s' € Ij, we have mssi = 2 and therefore ss' = s's. The following theorem shows that the study of Coxeter groups can be largely reduced to the case when F is connected. Theorem 1.15 Let (W, S) have Coxeter graph F, with connected components Fi, ... ,.Fr, and let Ii,... ,Ir be the corresponding subsets ofS. Then w = wh®---®wIr, and each Coxeter system (Wiv Ii) is irreducible. Proof. Since the elements of Ii commute with the elements of Ij, i ^ j, it is clear that the indicated parabolic subgroups centralize each other, hence that each is normal in W. Moreover, the product of these subgroups contains S and therefore must be all of W. According to theorem 1.14(c), for each 1 < i < r - 1, (Wh Wh... Wh) n W / i + 1 = {1}. It follows that W = Wh ® • • • 0 WIr (for example, see [Gal98]). . • Chapter 1. Basic Theory of Coxeter Groups 14 1.7 The Word and Conjugacy Problem Let a group G be given in terms of generators and relations. (i) For an arbitrary word w in the generators, decide in a finite number of steps whether w defines the identity element of G, or not. (ii) For two arbitrary words w\, w2 in the generators, decide in a finite number of steps, whether w\ and u>2 define conjugate elements of G, or not. The problems (i) and (ii) are called the word problem and the conjugacy problem, respectively, for the presentation defining G. It is shown in [Nov56], [Boo55] that there exist presentations of groups in which the word problem is not solvable, and there exist presentations of groups in which the conjugacy problem is not solvable [Nov54]. A very nice solution to the word problem for Coxeter groups was found by Tits [Tit69]. It allows one to transform an arbitrary product of generators from S into a reduced expression by making only the most obvious types of modifications coming from the defining relations. Here is a brief description. Let F be a free group on a set £ where £ is in bijection with S, and let 7r : F —> W be the resulting epimorphism. The monoid F+ generated by £ already maps onto W. If co € F+ is a product of various elements o € £, we can define l(co) to be the number of factors involved. If m = mst for s,t € S, the product of m factors of a and r; crra • • •, maps to the same element of W as the product of m factors TOT • • •. Replacement of one of them by the other inside a given LO G F+ is called an elementary simplification of the first kind; it leaves the length undisturbed. A second kind of elementary simplification reduces length, by omitting a consecutive pair aa. Write £(w) for the set of all elements of F+ obtainable from to by a sequence of elementary simplifications. Since no new elements of £ are introduced and length does not increase at each step, it is clear that £(w) is finite. It is also effectively computable. Clearly the image of £(w) under -n is a single element of W. Theorem 1.16 Let LO,LO' eF+. Then TT(CJ) = ir(u>') ijffE(w) n E(w') + 0. In particular, it(to) — lifflG £(u). . One direction is obvious. To go the other way, Tits assumes the contrary and analyses a minimal counterexample (in terms of lexicographic ordering Chapter 1. Basic Theory of Coxeter Groups 15 of pairs (ui, LO')): both elements must have the same length and T,(u>) consists of elements of equal length, etc., leading eventually to a contradiction. Much less seems to be known about the cdnjugacy problem for Coxeter groups. Appel and Schupp [AS83] have given a solution for extra large Cox-eter groups (those for which all mssi > 4 when s ^ s'.) 1.8 Finite Coxeter Groups In this section we restrict our attention to finite Coxeter groups. We will clas-sify all finite irreducible Coxeter groups in terms of their Coxeter graphs, in fact, we will give a complete list of all Coxeter graphs corresponding to finite irreducible Coxeter groups. According to theorem 1.15 every finite Coxeter group is isomorphic to a direct product of groups from this list. Recall in 1.3 the bilinear form B was not necessarily positive definite, the next theorem tells us that it is precisly when W is finite. Theorem 1.17 The following conditions on the Coxeter group W are equivalent: (a) W is finite. (b) The bilinear form B is positive definite. . ' The proof of this theorem is rather involved and so we refer the reader to Humphreys [Hum72]. If (W, S) is a Coxeter system with Coxeter graph Y (resp. Coxeter matrix M) then we say that Y (resp. M) is of finite-type if W is finite. Also, if the bilinear form B is positive definite then we call Y positive definite as well. Theorem 1.17 tells us that Y is positive definite if and only if it is of finite-type. Therefore, to classify the irreducible, finite Coxeter groups we just need to determine all connected, positive definite Coxeter graphs. Classification of all connected positive definite Coxeter graphs turns out -to be relatively straightforward. For a wonderful discussion and solution of the problem see Humphreys ([Hum72] sec. 2.3 - 2.7). It is shown in [Hum72] that the graphs in figure 1.1 are precisely all the connected positive definite Coxeter graphs. The letter beside each of the graphs in figure 1.1 is called the type of the Coxeter graph, and the subscript denotes the number of vertices. Recall ex-ample 1.2 shows the symmetric group on (n+l)-letters is a Coxeter group of typeA n . Chapter 1. Basic Theory of Coxeter Groups 16 An ( n > l ) Bn (n>2) Dn (n>4) E6 H3 l2{m) (m > 5) m 1 m m m m m m m m m 1 Figure 1.1: A l l the connected positive definite Coxeter graphs Chapter 1. Basic Theory of Coxeter Groups 17 Wr injects into Wp' r r An Am (for m > n), Bm (for m > n + 1), Dm (for m > n + 2), E8 (for n < 7), etc. B2 Bn (for n > 2), *4, I2(4) . B3 Bn (for n > 3), , ET, Eg E7 E8 HA H$, Hi Table 1.1: Inclusions among Coxeter groups The remarks after theorem 1.14 imply that if V is an induced subgraph of r ' then the corresponding Coxeter group Wv injects into Wv>. Table 1A lists some such inclusions for the Coxeter graphs in figure 1.1. Chapter 2 Basic Theory of Artin Groups The braid groups, which are the Artin groups of type An, were first intro-duced by Artin in [Art25], he further developed the theory in [Art47a,b] and [Art50], Since their introduction the braid groups have gone through a seri-ous line of investigation. One of the most influential papers on the subject was that of Garside [Gar69], in which he solved the word and conjugation prob-lems. Later, the connection of the braid groups with the fundamental group of a particular complex hyperplane arrangement lead to a natural general-ization: the Artin groups. In this chapter we introduce the Artin groups and discuss some of their basic theory. We follow closely the work of Brieskorn and Saito [BS72], which is a generalization of the work of Garside. 2.1 Definition Let M be a Coxeter matrix over S as described in section 1.1, and let Y be the corresponding Coxeter graph. Fix a set S in one-to-one correspondence with S. In the following we will often consider words beginning with a e S and in which only letters a and b occur, such a word of length q is denoted (ab)q so that (ab)q = aba ... q factors The Artin system of type Y (or M) is the pair (.4, E) where A is the group having presentation A = (a e E : •(ab)mab = (ba)mab if mab < oo). 18 Chapter 2. Basic Theory of Artin Groups 19 The group A is called the Artin group of type V (or M), and is sometimes denoted by Ar- So, similar to Coxeter systems, an Artin system is an Artin group with a prescribed set of generators. There is a natural map v : ^ 4r —• Wr sending generator a, £ £ to the cor-responding generator Si G S. This map is indeed a homomorphism since the equation (sjS 7) m^' = (sjSi)mii follows from sf = 1, = 1 and ( s j S j ) m ^ =• 1. Since z/ is clearly surjective it follows that the Coxeter group Wp is a quotient of the Artin group Ar- The kernel of v is called the pure Artin group, gener-alizing the definition of the pure braid group. From the observations in section 1.1 it follows that S is a minimal generating set for Ar- The homomorphism u has a natural set section r : Wr —> Ar defined as follows. Let w G W. We choose any reduced expression w = s\ • • • sr of w and we set T\W) = a\ • • • ar G Ar-By Tits' solution to the word problem for Coxeter groups (sec. 1.7), the defi-nition of T(W) does not depend on the choice of the reduced expression of w. Note that r is not a homomorphism. The Artin group of a finite-type Coxeter graph is called an Artin group of finite-type . In other words, .Ar is of finite-type if and only if the correspond-ing Coxeter group Wr is finite. A n Artin group Ar is called irreducible if the Coxeter graph F is connected. In particular, the Artin groups correspond-ing to the graphs in figure 1.1 are irreducible and of finite-type. These Artin groups are our main interest in the remaining chapters. 2.2 Positive A r t i n M o n o i d We now introduce the positive Artin monoid associated to the Artin system (A, E). A l l of the basic properties of Artin groups will follow from the study of the positive Artin monoid. Let 77s be the free group generated by E and F£ the free monoid generated by E inside F%; We call the elements of Fs words and the elements of F£ positive words. The positive words have unique representations as products of elements of E and the number of factors is the length I of a positive word. In the following we drop the subscript E when it is clear from the context. A n Chapter 2. Basic Theory of Artin Groups 20 elementary transformation of positive words is a transformation of the form U(ab)m"bV —> U(ba)mabV where U, V € F+ and a, b € S. A positive transformation of length t from a positive word U to a positive word V is a composition of t elementary trans-formations that begins with U and ends at V. Two words are positive equiv-alent if there is a positive transformation that takes one into the other. We indicate positive equivalence of U and V by U =p V. Note, it follows from the definition that positive equivalent words have the same length. We use = to denote equality in the group and = to express words which are equivalent letter by letter. The monoid of positive equivalence classes of positive words relative to T (or M) is called the positive Artin monoid (or just the Artin monoid) and is denoted A£. The natural map Ap —> .Ar is a homomorphism. We will see that for F of finite-type this map is injective. Recently, Paris [ParOl] has shown that for arbitrary Artin groups this map is injective. 2.3 Reduction Property The main result in this section concerns the positive Artin monoid and it ac-counts for most of the results we will encounter in this chapter. The statement is as follow. Lemma 2.1 (Reduction Property) For each Coxeter graph we have the following rule: If X and Y are positive words and a and b are letters such that aX =p bY then mab is finite and there exists a positive word U such that X =p (ba)mab-lU and Y =p (ab^^U. In other words, if aX =p bY then there is a positive transformation of the form a X —* > (ab)m«»U (ba)m"bU —> > bY taking aX to bY. The proof of this is long and tedious, we refer the reader to [BS72] for proof: . Chapter 2. Basic Theory of Artin Groups 21 A n analogous statement holds for reduction on the right side. We see this as follows. For each positive word U = ajj • • • ajfc define the positive word rev U by rev U = a i f c • • • , called the reverse or reversal of U. Clearly U —p V implies rev U =p rev V by the symmetry in the relations and the definition of elementary transforma-tion. It is clear that the application of rev to the words in lemma 2.1 gives the right-hand analog. It follows from the reduction propery that the positive Artin monoid is left and right cancellative. Theorem 2.2 If U, V and X, Y are positive words with UXV =p UYV then X=PY. -Proof. It suffices to show that left cancellativity holds since right cancellativ-ity follows by applying the reversal map rev . For U a word of length 1, say a, the reduction property implies that if aX =p aY then a word Z exists such that X =P {aa)m™-lZ = Z and Y =p {aa)maa~lZ = Z. Thus X —P Y. The result follows by induction on the length of U. ' • Let X, Y and Z be positve words. We say X divides Z (on the left) if Z = XY (if working in F+), Z =p XY (if working in A+), and write X\Z (interpreted in the context of F+ or A+). The term reduction property, which comes from [BS72], is appropriate as this property (in conjunction with left cancellativity) allows the problem of whether a letter divides a given word to be reduced to the same problem for a word of shorter length. In the following section we describe a method to determine when a given word is divisible by a given generator. Chapter 2. Basic Theory of Artin Groups 22 2.4 Divisibility Theory In this section we present an algorithm used to decide whether a given let-ter divides a positive word (in A+), and to determine the smallest common multiple of a letter and a word if it exists. 2.4.1 Chains Let a € E be a letter.' The simplest positive words which are not multiples of a are clearly those in which a does not appear, since a letter appearing in a word must appear in all positive equivalent words by the definition of ele-mentary transformation and the nature of the defining relations. Further, the words of the form (ba)q with q < mab are also not divisible by a. This follows from the reduction property. Of course many other quite simple words have this property, for example concatenations of the previous types of words in specific order, called a-chains, which we will now define. Let C be a non-empty word and let a and b be letters. We say C . i s a primitive a-chain with source a and target a if mac = 2 for all letters c in C. We call C an elementary a-chain if C = (6a)9 for some q < ma&. The source is a and the target is b if mab even and a if mab odd. An a-chain is a product C = C i • • • Cfc where for each i = 1,..., k, Ci is a primitive or elementary aj-chain for some a* G E , such that a\ = a and the target of C; is the source of Ci+i- This may be expressed as: C i t v c f c_i ck a = a\ ——> a,2 • 03 • •.• • afc ? afc+ 1 = o, The source of C is a and the target of C is the target of Ck- li this target is b then we say: C is a chain from a to b. Example 2.3 Let £ = {a, 6, c, d} and M be defined by mac = m ad = BIH = 2, ™-a& = rnbc = 3, m c d = 4. • • c, d, cd 2 c 7 are primitive a-chains with target a, • b, ba are elementary a-chains with targets a and b, respectively • a, ab, c, cb are elementary b-chains with targets b, a, b, c, respectively, The word ab^^cd^J)c^s&b_^ dec,sba_ C\ C2 C2, CA C 5 C§ Chapter 2. Basic Theory of Artin Groups 23 is a d-chain with target b, since C\ is a primitive d-chain with target d, C2 is an elementary d-chain with target c, C% is an elementry c-chain with target b, C4 is an elementary b-chain with target a, C 5 is a primitive a-chain with target a, and finally CQ is a simple a-chain with target b. The chain diagram for this example is: , Ci , C2 C3 , CA Cs Ce , a > d > c > 0 • •a > a - • 0. As the example 2.3 indicates there is a unique decomposition of a given a-chain into primitive and elementary factors if one demands that the primitive factors are a large as possible. The number of elementary factors is the length of the chain. Remark. If C is a chain from a to b then rev C is a chain from b to a. We have already noted that primitive and elementary a-chains are not divisible by a, the next lemma shows that this is also the case for a-chains. Lemma 2.4 Let C = C\ • • • Ck he a chain from a to b (where Ci is a primitive or elementary chain from a, to a^ +i for i = 1,..., k) and D is a positive word such that a divides CD. Then b divides D, and in particular a does not divide C. Proof. We prove this by induction on k. Suppose k = 1. Suppose C — x\ • • • xm is primitive, so maXi = 2 for all i. Then a; 1 • • • xmD =p aV for some positive word V. By the reduction property there exists a word U such that x2 • • • xmD =p (axi)maxi = all. Continuing in this way we get that a divides D, where a is the target of C. Supppose C = (ba)q is elementary, where mab > 2 and 0 < q < m0&. Then • (ba)qD=paV for some positive word V. By the reduction property, (ab)q~1 D=p(ab)ma'b~1U for some positive word U. So by cancellation, theorem 2.2, '•(ab)mab-qU if q is odd, {ba)mab~qU if q.is even. so D is divisible by a if q is odd, and b if q is even, which in each case is the target of C. This begins the induction. Suppose now k > 1. By the inductive hypothe-sis afc divides CkD, and by the base case, b = ak+i divides D. The last claim follows by taking D equal to the empty word. • D=p Chapter 2, Basic Theory of Artin Groups 24 Corollary 2.5 IfC is an a-chain such that a divides Cb, then b is the target ofC 2.4.2 Chain Operators Ka An. arbitrary word will in general not be an a-chain, for any particular a, and so we need to know firstly whether, given an arbitrary word U, there exists an a-chain C which is positive equivlent to U, and secondly how to calculate it and its target. We define operators Ka for each generator a which take as input a word U and output either • a word beginning with a if U is divisible by a, or • an a-chain equivalent to U if U is not divisible by a. Ka is called a chain operator (the K stands for Kette, German for chain). To state the precise definition of Ka, we need some preliminary definitions and notation. We call a primitive a-chain of length one or an elementary a-chain a simple a-chain, that is, a simple a-chain is a word of the form (ba}q where q < mab (where mab = 2 is allowed). For a simple a-chain of the form C = (ba)mab~1 we call C imminent and let C+ denote (ab)mab, so C+ =p Cc where c is the target of C. If D is any positive nonempty word denote by D~ the word obtained by deleting the first letter of D. For every letter a € S, we define a function Ka:F+—>F+ recursively. Let U be a word. If U is empty, begins with a or is a simple a-chain then . --Ka(U):=U. Otherwise, write U = CaDa where Ca and Da are non-empty words, and Ca is the largest prefix of W which is a simple a-chain, with target 6, say. The rest of the definition of Ka(U) is recursive on the lengths of U and Da: { CaK~b{Da) if Kb{Da) does not begin with b; or C+Kb{Da)~ if Ca imminent and Kb{Da) begins with 6; or Ka{CabKb{Da)-) otherwise Observe that Ka(U) is calculable. Chapter 2. Basic Theory of Artin Groups 25 Example 2.6 Computing Ka(U). Let S and M be as defined in example 2.3. First we will compute Ka of the word U = bcbabdc (notice U is not an a-chain). By the recursive nature of the definition ofKa we first need to decompose U as follows: U = ^ b^-^c^-^ba^- bdc^ Ci C2 C3 D where C\ is an a-chain with target a, C2 is an a-chain with target a, and C% is an a-chain with target b. Since D begins with the letter b then Kb(D) = D. Since C 3 is imminent, Ka(Cs • D) = C^D~ = abadc. Since C2 is imminent, and Ka(Cz • D) begins with the letter a, Ka(C2 • C3D) = C+ • Ka(C3 -D)-= ac • bade. Now Ka(C2CzD) begins with a but C\ is not imminent, so Ka(U) = Ka(Ci-C2CzD) = Ka(Ci • acbadc) since Ka(C2CzD) = acbadc = Ka(ba • cbadc) by definition of Ka. Applying the definition of Ka to the word bacbadc just returns the same word (try it!). Therefore, Ka(U) = bacbadc, which can be seen to be an a-chain positive equivalent to U, with target d. For our second example we will compute Ka of the word W = bacbacab. Again we need to decompose W as follows: W = ^ba^-^cb^-^a^-^cab^ Ci C2 C3 D, where C\ is an a-chain with target b, C2 is an b-chain with target c, and C 3 is a c-chain with target c. Since D begins with the letter c then KC(D). = D, so Kc(CzD) = C^D~ = ca • ab. Since C2 is imminent, KC(C2 • C 3 D ) = beb • aab. Finally, since C\ is imminent, Ka(W) = aba • cbaab. Chapter 2. Basic Theory of Artin Groups 26 Lemma 2.7 Let U be positive and a € S. Then (a) Ka(U) =p U and Ka(U) is either empty, begins with a or is an a-chain, (b) Ka{U) ~ U if and only ifU is empty, begins with a, or is an a-chain, (c) a divides U if and only ifKa(U) begins with a. ^ Proof, (a) If U is empty, begins with a or is a simple a-chain then Ka(U) = U and we are done. Otherwise, write U = CaDa where Ca and Da are nonempty and Ca is the longest prefix of U which is a simple a-chain. Let c denote the target of Ca. Since l(Da) < l(U) then by induction on length, KC(D) =p Da and KC(D) is either a c-chain or begins with c. If KC(D) is a c-chain then it cannot begin with c (lemma 2.4), so Ka{U) = CaKc(Ds) which is an a-chain, and moreover Ka(U) =p CaDa = U. Otherwise KC(D) begins with c. Considering first when Ca is imminent, we have Ka(U) = C+K 0 (D a )~ , which begins with a, and moreover, Ka{U)=pCacKc(Da)- = CaKc(D)=pCaDa = U. Otherwise se have Kc(Da) =p Da, Kc(Da) begins with c and Ca is not immi-nent; so Ka(U) = Ka(CacKc(Da)-). Now Cac is a simple a-chain of length greater than the length of Ca so by another induction, Ka(CacK0(Da)) begins with a or is an a-chain, and Ka{CacKc{Da)-) =p CacKc(Da)- = CaKc{Da) =p CaDa = U. (b) The direction (=>) follows from (a). To see the other direction notice the result is clear if U is empty, begins with a or is a simple a-chain. Suppose U is a nonempty a-chain, so U = CaDa where Ca is a simple a-chain with target c, say and Da is a c-chain. By induction since l(Da) < l(U), Kc(Da) = Da. Since Da is a c-chain it does not begin with the letter c thus by definition of -Ka, Ka(U) = CaKc(Da) = CaDa = U. (c) This follows from (a) and lemma 2.4 • Chapter 2. Basic Theory of Artin Groups 27 2.4.3 Division Algorithm Let U and V be words. We present an algorithm to determine whether U divides V (in A?) and in the case U divides V it returns the cofactor, i.e. the word X such that V =p UX. This can be done relatively easily using the chain operators Ka. Write U = a\ • • • a^. If U is to divide V then certainly a\ must divide V, this can be determined by calculating Kai (V) and checking if a\ is the first letter. If a\ is not the first letter then a\, and hence U, cannot divide V. Otherwise, we have Kai (V) = a\Kai (V)~. If U = a\ • • • were to divide V =p Kai (V) = a\Kai (V)~ then it is necessary for a2 to divide Kai (V)~. This can be determined by checking the first letter of Ka2(Kai(V)~). Continuing this way we either get that some aj does not divide Kai(Kai_1....Ka2(Kai(V)-)-•••)-) in which case U does not divide V, or d{ divides the above word for each 1 < i < k, in which case U divides V and the cof actor X is Kak'Kak_1..'-Ka2(Kai<V)-)----)-)-. We reformulate the above observations into the following definition. Let U and V be words. If U is empty then define (V : U) :=. V. Otherwise write U = Wa for some word W and some letter a. We make the recursive definition: ' - • • {co if (V : W) = oo, or if Ka{V : W) does not begin with a; or Ka(V : W)~ otherwise. Some remarks on the definition. 1 1. By induction of the length of U, if X is any word then (UX : U) = X. 2. Since Ka(X) is calculable for any word X, then (V : U) is also calcula-ble, for any pair of words V and U. Thus the following result gives a solution to the division problem in Ap. Lemma 2.8 The word U divides V precisely when (V : U) / oo, in which case V = p U(V : U). Chapter 2. Basic Theory of Artin Groups 28 Proof. If U is empty then the result clearly holds, so we may write U = Wa for some word W and some letter a. Suppose U divides V, so there is a word X such that UX = WaX =p V. By induction {V : W) + oo and V =p W(V : W). By cancellation, aX =p (V : W),so a divides (V : W). By lemma 2.7, ' Ka(V : W) begins with a, so (V :U) + oo and (V : U) =p X. On the other hand, suppose (V : U) ^ oo. Then (V : W) ^ oo, and in fact Ka(V : W) has to begin with a. By induction V =p W(V : W), so V = p W{V : W) =p WKa{V : W) =p WaKa{V : W)~ =p U{V : U) by the definition of (V : U). • Since we have a solution to the division problem in A? we get a solution to the word problem in A£ for free. Corollary 2.9 Tifo positive words U and V are positive equivalent precisely when (V : U) is the empty word. In section 2.6 we will show how to use this to solve the word problem in finite-type Artin groups Ap. 2.4.4 Common Multiples and Divisors Given a set of words Vi £ Ap where i runs over some indexing set I , a com-mon multiple of {Vi : i € 1} is a word U e A? such that every V divides U (on the left). A least common multiple is a common multiple which divides all other common multiples. If U and U' are both least common multiples then they divide one another, it follows by cancellativity and the fact that equivalent words have the same length that U =p U'. Thus, when a common multiple exists, it is unique. By a common divisor of {Vi : i G 1} we mean a word W which divides every Vj. A greatest common divisor of {Vi : i € 1} is a common divisor into which all other common divisors divide. Similarly, greatest common divisors, when they exist, are unique. With the help of the chain operators Ka defined in 2.4.2 we get a simple algorithm for producing a common multiple of a letter a and a word U, if one exists. The essence of the method lies in lemma 2.4 which can be rewritten to say: IfC is an a-chain to b, and U is a common multiple of a and C then U is a common < Chapter 2. Basic Theory of Artin Groups 29 multiple of a and Cb. Given an arbitrary word X, to calculate a common multiple with a generator a, we begin by applying Ka to X. If Ka(X) begins with a then we are done (X is divisible by a and so itself is a common multiple of a and X). Otherwise, Ka(X) is an a-chain, we determine its target b, and then concatenate it to get Ka(X)b = X'. If Ka(X') begins with a then we may stop; otherwise we repeat the process. If a common multiple exists, then the process will hault, producing a word which is in fact the least common multiple of a and X. Let a be a letter and W a word. The a-sequence of W is a sequence W 0 a , W f , . . . over F+ defined as follows. Set W 0 a := Ka(W), so by lemma 2.7, either WQ is empty, an a-chain or begins with a. Then for i > 1, define recur-sively {a if Wf_ 1 is empty; W/Li if W?_x begins with a; #a(W£_i&) if Wj 0 , ! is an a-chain to b. By lemma 2.7, W " is either an a-chain of begins with a (or if i = 0, may be empty). The a-sequence converges to a word W^ precisely when W£ begins with a. The following definition is intended to capture a notion of the limit of the a-sequence of W . W f c a i fW£ = W f c t t + 1;-or co otherwise. The following example illustrates the way in which L(a, W) = oo Example 2.10 Let S = {a, b, c} and M , f/ze Coxeter matrix, he defined by mab = mac = mbc = 3. (Note, by the results in 1.8 Ar is not of finite type.) Consider the word W = be. Observe that for any k > 1, Uf. = (bacbac)k is an a-chain with target a. The first member of the a-sequence ofW is WQ = be = Uobc, and then for all k>0, W6k+2 = Ukbaca, W 6 a f c + 5 = Ukbacbabc, Thus, the a-sequence never L(a, W) :=• W 6 a f c = Ukbc, • W 6 a f c + 1 = C/fe6ca, wSk+3 = Ukbacab, W 6 a f c + 4 = Ukbacbab, and so W g f c + 6 = Ukbacbacbc = Uk+ibc and so on. converges to a word, and so L(a, be) = oo. / Chapter 2. Basic Theory of Artin Groups 30 The following result characterizes the situation when L(a, W) ^ oo. Lemma 2.11 L(a, W) ^ oo precisely when a and W have a common multiple,, in which case L(a, W) is a least common multiple of a and W begins with a. • Proof. If W is empty then Wg =• W and Wf = a for all? > 1. Thus L(a, W) = a, and so the result holds trivially. So we may that suppose W is nonempty. Suppose that a and W have a common multiple M. By lemma 2.7, we know that W Q = Ka(W) —p W and so divides M. Since. W is nonempty, W§ either begins with a or is an a-chain, is a multiple of W and divides M. We will show that the same is true of all Wf, using induction on i. Suppose that, for a given i > 0, Wf is a multiple of W and divides M. If Wj" begins with a, then Wj1 = Wf for all j > i, and so we are done. Otherwise, Wf is an a-chain to b and, by lemma 2.4, M is a common multiple of W?b =p Ka(Wfb) = W°-+l and a. Since W divides Wf then W must also divide Thus we have shown that when a and W have a common multiple, every element of the a-sequence of W is a multiple of W, and divides M . Since elements of the a-sequence increase in length until an element begins with a, and since divisors of M cannot exceed M in length, eventually there is a first W% which begins with a. Hence L(a, W) = W%. Futhermore, we have shown that L(a, W) divides every common multiple M of a and W, making it a least common multiple. On the other hand, suppose L(a, W) ^ oo. Then there is a first number k > 0 such that W% begins with a. If k = 0, then L(a, W) = Wg =p W. If k > 0 then by definition of the a-sequence, there are letters bi,...,bk which are targets of the a-chains WQ , . . . , W%_v respectively, and for each i < k, Wzabi+1 =p Wf + 1 , so L(a, W) = W£ =p Wgbx---bk.=p Wbx---bk. hence L(a, W) is a common multiple of a and W. • Thus we have-in L(a, W) a calculator of least common multiples of a gen-erator and a word. By repeated application of this operation, we can obtain least common multiples of arbitrary pairs of words. Chapter 2. Basic Theory of Artin Groups 31 Let V and W be words. Define recursively: UV, W) := W if V is empty; or aL(U, L(a, W)~) if V = aU,L(a, W) ± oo and L(U,L(a,W)-) 7^.00; or 00 otherwise. Similar to lemma 2.11 we get the following lemma. Lemma 2.12 L(V, W) ^ 00 -precisely when V and W have a common multiple, in which case L(V, W) begins with V and is a least common multiple of V and W. Moreover, L(V, W) / 00 precisely when L(W, V) ^ 00, in which case L(V, W) =p L(W,V). We can also compute the least common multiple of any finite collection of words by induction on the number of words. In particular, let V i , . . . , Vm be words and let 1 denote the empty word. Define recursively: L ( V i , . . . , V r o ) := { 1 m. = 0; or Vi if m = 1; or 00 m > 2 and Liy2, • ••, Vm) = 00; or L(Vi, L(V2,..., Vm)) if m > 2 and L(V2,..., Vm) + 00. The next result follows by induction on m using lemma 2.12. Lemma 2.13 L ( V i , . . . , Vm) ^ 00 precisely when V i , . . . , Vm have a common mul-tiple, in which case L(V\,..., Vm) begins with Vi and is a least common multiple of V i , . . . , Vm. Moreover, for any permutation a of {1,... m}, L ( V i , . . . , Vm) ^ 00 if and only if L(V r C T( 1),. . . , V a(m)) / 00, in which case L(V\,..., Vm) =p Corollary 2.14 Let fl be a finite set of words. Then Q has a common multiple if and only if it has a least common multiple. • Since E is finite then an infinite set of words in F+ must have elements of arbitrary length. Since positive equivalent words have the same length it Chapter 2. Basic Theory of Artin Groups 32 follows that a common multiple must be at least as long as any of the factors. So an infinite set of words can have no common multiples. On the other hand, the empty word divides every other word, so an arbitrary nonempty set fl of words has a common divisor. If D denotes the set of all common divisors of fl, then D is finite by the preceding discussion. Since every element of fl is a comon multiple of D, then by corollary 2.14, D has a least common multiple, which is a greastest common divisor of fl. Thus, greatest common divisors for nonempty sets of words always exist. Remark. The only letters arising in the greatest common divisor and the least common multiple of a set of words are those occurring in the words them-selves. Proof. For the greatest common divisor it is clear, because in any pair of positive words exactly the same letters occur. For the least common multiple, recall how we found L ( a , W): W$. = Ka(W), and W?+1 = W? if W? starts with a, or W?+1 = Ka(W£b) if W f is an a-chain from a to b. But if b ^ a, then the only way we can have an a-chain from a to b is if there is an elementary subchain somewhere in the a-chain containing 6. So W?+l only contain letters which are already in W f . • 2.4.5 Square-Free Positive Words When a positive word U is of the form U = XaaY where X and Y are posi-tive words and d is a letter then we say U has a quadratic factor. A word is square-free relative to a Coxeter graph T when U is not positive equivalent to a word with a quadratic factor. The image of a square-free word in Ap" is called square-free. Lemma 2.15 Let V be a positive word which is divisible by a and contains a square. Then there is a positive word V with V =p V which contains a square and which •begins with a. Thus, if W is a square-free positive word and a is a letter such that aW is not square free then a divides W. Proof. The proof is by induction on the length of V. Decompose V, as V = Ca(V)Da(V) Chapter 2. Basic Theory of Artin Groups 33 where Ca(V) and Da(V) are non-empty words, and Ca{V) is the largest prefix of V which is a simple a-chain. Without loss of generality we may assume that V is a representative of its positive equivalence class which contains a square and is such that l(Ca{V)) is maximal. When Ca(V) is the empty word it follows naturally that V = V satisfies the conditions for V. For nonempty Ca(V) we have two cases: (i) Da(V) contains a square. By the induction assumption, one can assume, without loss of generality that Da(V) begins with the target of the simple a-chain Ca(V). Thus, since the length of Ca(V) is maximal, Ca(V) is of the form (ba)mab~1. From this it follows that when Da(V)~ contains a square then V = aCa(V)Da(V)~ satisfies the conditions for V, and otherwise V = a?Ca{V)Da(V)— does. (ii) Neither Ca(V) nox-Da{V) contains a square. Then V is of the form V = (ba)qDa(y) where q > 1, and Da(V) begins with a if q is even, and b if q is odd. If q even then (ba)q. is a simple a-chain with target b so, by lemma 2.4, since a divides (ba)qDa(V), b divide Da(V). But Da(V) begins with a so by an application of the reduction property there exists E such that Da(V)=p.(ba)m«bE. Similarly, for q odd. Then V = a ( 6 a ) m o t _ 1 ( 6 o ) 9 £ ; if m a 6 is even, V = a(ba)mab-l{ab)qE i fm a 6isodd. satisfies the conditions. To prove the second statement, we have that there exists a positive word U, such that all contains a square and aW =p all from the first statement. It follow from cancellativity that U =p W and, since W is square free, that U does not contain a square. So U begins with a and W is divisible by a. • From this lemma we get the following result concerning the a-sequence of a square-free word W, which will be needed in the next section. Lemma 2.16 If W is a square-free positive word and a is a letter then each word Wf in the a-sequence of W is also square-free. Chapter 2. Basic Theory of Artin Groups 34 Proof. W§ is square-free since W§ =p W. Assume W" is square-free. Then either W?+l = Wf or W?+i =P Wfbi where h is the target of the chain Wf. If W^bi is not square-free then fyrev W f is not square-free and by lemma 2.15, the 6i-chain rev W? is divisible by 6*, in contradiction to lemma 2.4. • Let QFAp be the set of square-free elements of A.p~. Consider the canon-ical map of QFAp into the Coxeter group Wr defined by the composition of the canonical maps Ap~ —> Ar —-> Wr- It follows from theorem 3 of Tits [Tit69] that QFAf —> Wr is bijective. Thus, QFAf is finite precisely when Ar is of finite type (i.e. Wr is finite). This result is needed in the next section. 2.5 The Fundamental Element Let M be a Coxeter matrix over E, and let I c E such that the letters of I in «4p" have a common multiple. Then the uniquely determined least common multiple (which exists by lemma 2.13) of the letters of I in Ap" is called the fundamental element Aj for I G Ap. The word "fundamental", introduced by Garside [Gar69], refers to the fundamental role which these elements play. It is shown in [BS72] that when Ar is irreducible (i.e. T connected) and there exists a fundamental element A s , then A s or A | generates the center of .Ar- The conditions for the exis-tence of A s are very strong and are outlined in the following two theorems, which appear in [BS72]. . Theorem 2.17 For a Coxeter graph T the following statements are equivalent: (i) There is a fundamental element A s in Ap~. (ii) Every finite subset ofAp has a least common multiple. (iii) The canonical map Ap~ —• .Ar is injective, and for each Z G Ar there exist X,Y G A+ with Z = XY~\ -(iv) The canonical map Ap —> A r is injective, and for each Z e Ar there exist X, Y G Ap~ with Z — XY'1, where the image ofY lies in the center of Ar-Theorem 2.18 Let The a Coxeter graph. There exists a fundamental element A s in Ap if and only ifT is of finite-type (i.e. Wr is finite). Chapter 2. Basic Theory of Artin Groups 35 To prove theorem 2.18 we need to recall the theorem of Tits we discussed at the end of section 2.4.5 on page 34: F is of finite-type if and only.if QFAp is finite. It is shown in [BS72] that every element of QFAp divides A s thus if A s exists then QFAp must be finite. To prove the converse suppose that A s does not exist in Ap. Let J = { a i , . . . , afc} C E be such that A j exists but A J U { a f c + 1 } does not exist (here we have assumed £ has been ordered). Then the afc+i-sequence of A j does not terminate. Since A j is square-free (see [BS72]) then by lemma 2.16 every element of the ak+i-sequence of A j is square free (and distinct). Thus QFAy is infinite. It is important to note that in theorem 2.17 the positive words X and Y such that Z = XY~X are calculable. This can be seen from the proof given in [BS72]. We use this fact in 2.6 to solve the word problem for finite-type Artin groups. 1 For a complete discussion on properties of the fundamental element see [BS72]. There it is shown that the image of the fundamental element of Ap in the Coxeter group Wr is precisely the longest element. Also they give formu-lae for the fundamental elements of irreducible finite-type Artin groups, i.e. the Artin groups corresponding to the Coxeter graphs in figure 1.1. 2.6 The Word and Conjugacy Problem In this section we use the machinary developed thus far to give a quick solu-tion to the word problem for finite-type Artin groups. The conjugacy problem is also discussed. Let U, V G Ar, where F is of finite-type. We want to decide if U = V. By theorem 2.17 we know there exists (calculable) positive words Xi,X2, Y%, Y2 G Ap such that U = X1Y1~1 and V = X2Y2~X where the images of Y\ and Y2 are central in Ar- To decide ii U — V it is equivalent to decide if X{Y2 = X2Y\, but since the canonical map Ap —> Ar is injective this is equivalent to deciding if X{Y2 —p X2Y\. In 2.4.3 we gave a solution to the word problem for Ap, thus a solution to the word problem for ,4r follows. In [BS72] it is shown elements oi Ap and .Ar can be put into a normal form using the fundamental element. This also gives a solution to the word Chapter 2. Basic Theory of Artin Groups 3.6 problem in both A^ and Ar- Brieskorn and Saito also give a solution to the conjugacy problem in finite type Artin groups. Another solution to the word and conjugacy problems appears in [Cha92]. It is shown that finite-type Artin groups are biautomatic in which case they are known to have solvable word and conjugacy problems. Some infinite-type Artin groups have been shown to have solvable word and conjugacy problems. Appel and Schupp [AS83] solve these problems for Artin groups of extra-large type (i.e. mab > 4 for all a, b G £). 2.7 Parabolic Subgroups Let (Ar, S) be an Artin system with values mab for a, b £ £. For a subset I c £ we define A r 7 to be the subgroup of Ar generated by I. We call the subgroup Arj a parabolic subgroup. (More generally, we refer to any conjugate of such a subgroup as a parabolic subgroup.) Van der Lek [Lek83] has shown that for each I c £ the pair (Arj,I) is an Artin system associated with T/. That is, parabolic subgroups of Artin groups are indeed Artin groups. A proof of this fact also appears in [Pa97]. Thus the inclusions among Coxeter groups in table 1.1 also hold for the associated Artin groups: Crisp [Cri99] shows quite a few more inclusions hold among the irreducible finite-type Artin groups. Table 2.1 lists these inclusions. Notice that every irreducible finife-type Artin group embeds into an Artin group of type A, D or E. Similar to that of Coxeter groups we have that the study of Artin groups can be largely reduced to the case when V is connected. Theorem 2.19 Let (Ar, £ ) have Coxeter graph Y, with connected components Y\, ..., YT, and let I\,... ,Ir be the corresponding subsets o/£ . Then Ar = Arh ®---®ATlr, and each Artin system (ArT., W is irreducible. Cohen and Wales [CW01] use this fact and the fact that irreducible finite type Artin groups embed into an Artin group of type A, D or E to show all Artin groups of finite-type are linear (have a faithful linear representation) by showing Artin groups of type D, and E are linear, thus generalizing the recent result that the braid groups (Artin groups of type A) are linear [BiOl], [Kr02]. Chapter 2. Basic Theory of Artin Groups 37 Ar injects into Ar> r r An Am (m > n), ' Bn+i {n > 2), Dn+2, Ee(l<n< 5), E7 (1 < n < 6), £ 8 ( l < n < 7), FA, H3{l<n< 2), HA (1 < n < 3) I 2 ( 3 ) (1 < n < 2) Bn An, A2n-\, A2n, Ee E-j, E% E7 E% . F4 E&, E-J, E% H3 D& HA Eg h(m) Am—l Table 2.1: Inclusions among Artin groups Chapter 2. Basic Theory of Artin Groups 38 2.8 Geometric Realization of Artin Groups In this section we discuss how finite-type Artin groups appear as fundamen-tal groups of complex hyperplane arrangements. From this point of view we can see that finite-type Artin groups are torsion free. Let (Wr, S) be a Coxeter system where Wr is finite and \S\ = n. Let V be the associated (real) n-dimensional vector space, and B the bilinear form on V introduced in section 1.3. We know from theorem 1.17 that V is a Euclidean space. Let T denote the set of reflections in W. For each t £ T let Ht denote the hyperplane in V (pointwise) fixed by t. Let Ti = {Ht}teT be the collection of such hyperplanes. The complement of Ti in V is defined by M(H).= V\ | J H. Hen Note that since V is a real vector space M(Ti) is not connected. However, if we "complexify" V and the arrangement of hyperplanes Ti we get a connected space. This is done as follows. The complexification of V is Vc ,= C N . The complexification of a hyperplane H is the hyperplane He of Vc having the same equation as H. The complexification of Ti is the arrangement Tic = {He '• H G Ti.} in Vc- The topological space M(HC) = VC\ | J H. is our primary interest. Before we proceed any futher we need to make some definitions. A collec-tion of hyperplanes Ti in a (real) vector space is called a (real) arrangement of hyperplanes. We say Ti is central if all the hyperplanes of Ti contain the ori-gin. We say further that Ti is essential if the intersection of all the elements of Ti is {0}. Call Ti simplicial if it is central and essential, and if all the.chambers of Ti (i.e. connected components of V \ \JHen H) are cones over simplices. The following theorem indicates the importance of knowing an arrangement is simplicial. Theorem 2.20 (Deligne [Del72]). Let Ti be a simplicial arrangement of hyper-planes. Then M(Hc) is an Eilenberg-Maclane space (i.e. its universal cover is con-tractible). Chapter 2. Basic Theory of Artin Groups 39 The importance of this theorem lies in the fact that if M(G) is a finite dimen-sional Eilenberg-Maclean space for a group G then G has finite cohomological dimension and so, from a result in homological algebra, G is torsion-free. Let us return now to our particular hyperplane arrangement H defined above. It follows from our work in chapter 1 that the arrangement of hy-perplanes H = {Ht}teT is central and essential. Futhermore, Deligne [Del72] showed that H is simplicial. Thus, it follows from theorem 2.20 that M(Hc) is an Eilenberg-Maclean space. Deligne has shown that the fundamental group of M(Hc) is precisely the pure Artin group associated with T. Moreover, Deligne showed that Wr acts freely on M(Hc) so that M(Hc)/Wr is also an Eilenberg-Maclean space and -Ki(M(T-L<c)/Wr) is the Artin group Ar- Thus, Ar is torsion-free. For arbitrary Artin groups Ar (not necessarily of finite-type) more general constructions of K(Ar, l)-spaces have been done, for example see [CD95]. A n algebraic argument showing finite-type Artin groups are torsion free was discovered by Dehornoy [Deh98]. The proof uses the divisibility theory we developed in this chapter. Chapter 3 Commutator Subgroups of Finite-Type Artin Groups Gorin and Lin [GL69] gave a presentation for the commutator subgroup 23^ of the braid group 5 B n , n > 3 , which showed 9 3 ^ is finitely generated and perfect for n > 5 . This has some interesting consequences concerning Q 3 n and "orderability", which we discuss in chapter 5. In this chapter we extend the work, of Gorin and Lin and compute presentations for the commutator subgroups of all the other irreducible finite-type Artin groups; those corre-sponding to the Coxeter graphs in figure 1.1. This will be applied in chapter 4 to "local indicability" of finite-type Artin groups. 3.1 Reidemeister-Schreier Method We will use the Reidemeister-Schreier method to compute the presentation for the commutator subgroups so we give a brief overview of this method in this section. For a complete discussion of the Reidemeister-Schreier method see [MKS76]. Let G be an arbitrary group with presentation (ai,... J an : i?M(a v), . . .) and H a subgroup of G. A system of words TZ in the generators ai,...,an is called a Schreier system for G modulo H if (i) every right coset of H in G contains exactly one word of TZ (i.e. TZ forms a system of right coset repre-sentatives), (ii) for each word in TZ any initial segment is also in TZ (i.e. initial segments of right coset representatives are again right coset representatives). Such a Schreier system always exists, see for example [MKS76]. Suppose now that we have fixed a Schreier system TZ. For each word W in the generators 40 / Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 41 a i , . . . , a n we let W denote the unique representative in TZ of the right coset HW. Denote SK,av = Kav • Kav \ (3.1) for each K e TZ and generator av. A theorem of Reidemeister-Schreier (theo-rem 2.9 in [MKS76]) states that H has presentation (SK,av,----SM,ax,---',T(KRIJ,K~1),...) (3.2) where K is an arbitrary Schreier representative, av is an arbitrary generator and R^ is an arbitrary defining relator in the presentation of G, and M is a Schreier representative and ax a generator such that Ma\ ~ Ma\, where « means "freely equal", i.e. equal in the free group generated by {ai,..., an}. The function r is a Reidemeister rewriting function and is de-fined according to the rule r(a?ei • ••a'") = s% n • '• • s% „ (3.3) v n iPJ K j j ^ j j Kip,aip \ ' where Ki. = a-1 • • • a6/"1, if e, = 1, and ifj, = a-,1 • • • ae/, if e, = - 1 . It should be noted that computation of T(U) can be carried out by replacing a symbol al of U by the appropriate s-symbol seK a v . The main property of a Reide-meister rewriting function is that for an element U € H given in terms of the generators a„ the word T(U) is the same element of H rewritten in terms of the generators SK,O,V 3.2 A Characterization of the Commutator Subgroups The commutator subgroup G' of a group G is the subgroup generated by the elements [51,52] := 9i929i1921 r o r a n 9i>92 £ :G. Such elements are called commutators. It is an elementary fact in group theory that G' is a normal subgroup in G and the quotient group GjG' is abelian. In fact, for any normal subgroup N < G the quotient group G/N is abelian if and only if G' < N. If G Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 42 is given in terms of a presentation (Q : K) where Q is a set of generators and 1Z is a set of relations, then a presentation for G/G' is obtained by obelianizing the presentation for G, that is, by adding relations gh = hg for all g,h G Q. This is denoted by (Q : TZ) A b . Let U G Ar, and write U — a-1 ••• • a^, where = ± 1 . The degree of U is defined to be Since each defining relator in the presentation for Ar has degree equal to zero the map deg is a well defined homomorphism from Ar into Z. Let Zr denote the kernel of deg; Zr = {U € Ar •• deg(U) = 0}. It is a well known fact that for the braid group (i.e. F = An) ZAK is precisely the commutator subgroup. In this section we generalize this fact for all Artin groups. Let ToM denote the graph obtained from T by removing all the even-labelled edges and the edges labelled oo. The following theorem tells us ex-actly when the commutator subgroup A'r is equal to Zr-Theorem 3.1 For an Artin group Ar, F0dd is connected if and only if the commu-tator subgroup Ar is equal to Zr-Proof. For the direction (=>) the hypothesis implies Ar/A'y ~ Z. Indeed, start with any generator ai, for any other generator a,j there is a path from <2j to aj in F0(id-ai = a i i * a i 2 * ' ' ' > a i m = aj • Since mi f c i f c + 1 is odd the relation K ^ + 1 )m ^ f c + 1 = ( a l k + 1 a i k ) m ^ . becomes atk = a,ik+1 in Ar/A'r- H e n c e / a% = aj m Ar/A'T. It follows that, Ar/A'r ^ (a i , . . . , an : a x = • • • = an) ~ Z, where the isomorphism 4>: Ar/Ar —> Z is given by Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 43-UA'r^deg(U). Therefore, A'r = ker^ = Zr- ' We leave the proof of the other direction to theorem 3.2, where a more general result is stated. . • For the case when F0dd is not connected we can get a more general de-scription of A'r as follows. Let F0dd have m connected components; F0dd = Ti U . . . U T m . Let Ej c S be the corresponding sets of vertices. For each 1 < k < m define the map degfc : Ar —> Z as follows: If U = a-1 • • • a- r G Ar take degfc(C7) =  er l<j<r where ai^E^k It is straight forward to check that for each 1 < k < m the map degfc agrees on (ab)rnab and (ba)mab for all a, b G E . Hence, degfc : Ar —>• Z is a homomor-phism for each 1 < k < m. Let Zr{m):= f ) ker(degk). \<k<m The following theorem tells us that this is precisely the commutator subgroup of -4[ -Theorem 3.2 Let The a Coxeter graph such that TQdd has m connected components. Then A'T = Z r { m ) . Proof. Clearly A'T C Zr^ since commutators certainly lie in the kernel of degfc for each k. To show the opposite inclusion let W G Zr^m\ i.e. deg f c(W) = 0 for all 1 < k < m. Since . Ar/AT ~ {ai,...,an: (aiaj)ma^ = (ajai)ma^)Ab ~ {ai,... ,an : cn = aj iff i and j lie in the same connected component of r o ^ ) , ~ Z ' " . with the isomorphism given by Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 44 U A T ^ (deg1(C/),...,degm(c/)) / . then WA'r must be the identity in Ar/Ar (since it is in the kernel). In which case W € A'r. • It is this characterization of the commutator subgroup which allows us to . use the Reidemeister-Schreier method to compute its presentation. In partic-ular, we can find a relatively simple set of Schreier right coset representatives. 3.3 Computing the Presentations In this section we compute presentations for the commutator subgroups of the irreducible finite-type Artin groups. We will show that, for the most part, the commutator subgroups are finitely generated and perfect (equal to its com-mutator subgroup). Figure 3.1 shows that each irreducible finite-type Artin group falls into one of two classes; (i) those in which Todd is connected and (ii) those in which T0dd has two components. Within a given class the arguments are quite sim-ilar. Thus, we will only show the complete details of the computations for types A n and Bn. The rest of the types have similar computations. 3.3.1 Two Lemmas We will encounter two sets of relations quite often in our computations and it will be necessary to replace them with sets of simpler but equivalent re-lations. In this section we give two lemmas which allow us to make these replacements. Let {pk}kez > a> b, and q be letters. In the following keep in mind that the relators Pk+iPk+2Pk1 s P n t U P i n t 0 t n e t w o tyPes °f relations Pk+2 = Pk~lPk+i (for k > 0), and pk = Pk+iPk+2 ^ o r k <0). The two lemmas are: Lemma 3.3 The set of relations Pk+\Pkl2Pkl = 1' PkaPk+2a~1Pkl1a~l = 1, b = p0apQ1, (3.4) Chapter 3. i Commutator Subgroups of Finite-Type Artin Groups 45 (An)odd (n > 1) • ai 0,2 03 O-n-2 ttn-1 &n (Bn)odd (n>2) • ai a 2 (13 • • • dn-2 O.n-1 On (Dn)odd ( n > 4 ) • 6) odd (3 7) odd odd a\ 02 03 B n - l 1a 6 ai 02 (13 04 as > • • a\ 0,2 03 14 as 16 1a 8 • • • • a\ a2 03 04 .as a6 .07 an-3 , a„_2 [Ft) odd {Hz) odd (H4) odd (h(m))odd {m > 5) ai a 2 03 04 ai a 2 a3 ai a 2 a3 04 ai a 2 ai 02 m odd m even Figure 3.1: Fodd for the irreducible finite-type Coxeter graphs F Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 46 is equivalent to the set PobPo1 = PiapT1 = PibpV1- = 1, b, b2a-lb (a^bfarH. pkq =,qpk+1, qpi, Piq = qPo 1Pi-(3.5) (3.6) (3.7) (3.8) (3.9) Lemma 3.4 The set of relations: Pk+iPlliPl1 = 1, is equivalent to the set Pk+iPkLPk1 = l> PM = The proof of lemma 3.4 is straightforward. On the other hand, the proof of the lemma 3.3 is somewhat long and tedious. Proof. [Lemma 3.4] Clearly the second set of relations follows from the first set of relations since p2 .= PQ lpi- To prove the converse we first prove that pkq = qpk+\ (k > 0) follows from the second set of relations by induction on k. It is easy to see then that the same is true for k < 0. For k = 0,1 the result clearly holds. Now, for k = m + 2; Pm+2qPm+3q~1 = Pm+2qpm\2Pm+iq~l, = Pm+2{pm\iq)Pm+iq~l by IH (k = m + 1), = Pm+2Pm1+i(qpm+i)q~l, . = Pm+2Pm+1(pmq)q~1 b y I H ( /c = m ) , = Pm+2Pm1+\Pm, = 1-• Proof. [Lemma 3.3] First we show the second set of relations follows from the first set. Taking k = 0 in the second relation in (3.4) we get the relation p0ap2a~lpi1a~1 = 1, Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 47 and, using the relations p2 = p0 lpi and b = poap0 \ (3.8) easily follows. Tak-ing k = 1 in the second relation in (3.4) we get the relation Piap3a~lp2 1a~1 = 1. Using the relations p% = p1~1p2 and p2 = p^Pi this becomes Piap^p^pia^p^poa'1 = 1. But piapj1 = a~xb (by (3.8)) so this reduces to a~lbpQlb~1apoa~l =. 1. Isolating bp^ 1 on one side of the equation gives bpQ 1 = O?PQ 1a~1b. r Multiplying both sides on the left by po.and using the relation poap^1 = b it easily follows pobpg 1 = b2a~lb, which is (3.7). Finally taking k = 2 in the second relation in (3.4) we get the relation P2ap4a~1p^1a~1 = 1. Using the relation p4 =-p2~1P3 this becomes P2ap~2~ 1P3a~lP3 loTl = 1. (3.10) Note that P^Piap^Po byp2=p0~1pi p^a^bpo by (3.8) a~2ba'la using (3.4) and (3.7) a~2b Pi1p2ap21Pi byp 3 = p 1 " 1 p2 . Pila-2bpu p2ap21 = and p-sapz1 = Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 48 where the second equality follows from the previous statement. Thus, (3.10) becomes a~2bpV1b~1a2pia~1 = 1 Isolating the factor bpV1 on one side of the equation, multiplying both sides by pi, and using the relation (3.8) we easily get the relation (3.9). Therefore we have that the second set of relations (3.5)-(3.9) follows from the first set of relations (3.4). In order to show the relations in (3.4) follow from the relations in (3.5)-(3.9) it suffices to just show that the second relation in (3.4) follows from the relations in (3.5)-(3.9). To do this we need the following fact: The relations pkapll = akb, (3.11) ' p.bp-1 = (a'kb)k+2a^k+1\ (3.12) p~\Pk = ab-'a^2, (3.13) p-^bpk = (ab-lak+2)ka, (3.14) follow from the relations in (3.5)-(3.9). The proof of this fact is left to lemma 3.5 below. From the relations (3.ll)-(3.14) we obtain Pfc+iaPfc+i = a - ( f c + 1 ) 6 = a'1 • a~kb = a^pkapl1, (3.15) and Pk+iaPk+i — a b _ 1 a f c + 3 = ab~1ak+2a = pi lapka. (3.16) Now we are in a position to show that that the second relation in (3.4) follows from the relations in (3.5)-(3.9). For k > 0 PkdPk+2a~1Pkl1a~l = Pkap^Pk+ia^Pkli^1 by (3.5) v v ' = PkapF1(a~1pkapI1)~1a~1 by (3.15) = 1. and for k < 0 Pkapk+2a~lPklia~1 = Pk+iPkl2aPk+2a~1pllla'1 by (3.5) = Pk+i(Pkliapk+ia)a'1Pklia~1 by (3.16) = 1. Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 49 Therefore, the relations Pkapk+2a~1Pkl1a~1 = 1, k 6 Z follow from the relations in (3.5)-(3.9). • To complete the proof of lemma 3.3 we need to prove the following. Lemma 3.5 The relations Pkapkl = akb pkbp^ = (a-kb)k+2a-(k+% p^apk = a & - 1 a f c + 2 pfbpk = {ab-lak+2)ka follow from the relations in (3.5)-(3.9). Proof. We will use induction to prove the result for nonnegative indices k, the result for negative indices k is similar. Clearly this holds for k = 0 , 1 . For k = m + 2 we have P m + 2 a p ~ + 2 = PmPm+iaPm+lPrn by (3.5), = pZ^a~^m+1^bpm by induction hypothesis (IH), = {pmla-^pm){pmlbpm), . = {Pmlapmr{m+l\pmlbpm), = \ a b - l a m + 2 ) ^ m + l ) ( a b - l a m + 2 ) m a by IH, = \ab-1am+2)-1a, • = a^"H2)b. Pm+2bpm\2 = PmPm+lbp'^Pm by (3.5), = p - 1 ( a - ( m + 1 ) 6 ) m + 3 a - ( r a + 2 ) 6 p m by IH, = ( ( P ^ a ^ ) ^ ^ 1 ) ^ - 1 ^ ) ) ^ 3 ^ - ^ ^ , , , ) - ^ 2 ) ^ - 1 ^ , = ( (ar 1 a r a + 2 ) - ( m + 1 ) (ar 1 a m f l ) m a) ( , n + 3 I • ( a 6 - 1 a m + 2 ) - ( m + 2 ) ( a 6 - 1 a T n + 2 ) m a by IH, = ( a - ^ b ) m + 3 ( a b - l a m + 2 ) - \ = (a-(m+2h)m+Aa-(m+3h; Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 50 Similarly for the other two equations. Thus, the result follows by induction. • 3,3.2 Type A The first presentation for the commutator subgroup *&'N+X = A'AN of the braid group 25 n + i = A A „ appeared in [GL69] but the details of the computation were minimal. Here we fill in the details of Gorin and Lin's computation. The presentation of A A U is AA„ = {ai, —,an : ataj = a^a* for | i —. j | > 2, ajaj+iaj = a^iaia^i for 1 < i < n — 1 ). Since (An)0dd is connected then by theorem 3.1 A'AN = ZA„. To simplify no-tation in the following let Zn denote A'AN = ZAn- Elements U, V G A A U lie in the same right coset precisely when they have the same degree: znu = znv uv-'eZn • . • < = > deg(f/) = deg(y>, thus a Schreier system of right coset representatives for A A „ modulo Zn is K = {a\ : / c £ Z } By the Reidemeister-Schreier method, in particular equation (3.2), Zn has gen-erators sak a . := ^^(a^aj)-1 with presentation <sa*.«i' • • • : S*?A» • • •' r(a{Ria7e), • • •, T(a[Titja7e),...), (3.17) where j € {1 , . . .,n},k,£ £ Z; and m G Z, A G { 1 , . . . ,n} such that af-ax « ama\ ("freely equal"), and Ty , Ri represent the relators aiCLja^aJ1, > 2, and aiai+iaia'^aT1 a~+v respectively. Our goal is to clean up this presenta-tion. The first thing to notice is that a^ax « HfaTx = a ™ + 1 A = 1 Thus, the first type of relation in (3.17) is precisly saf,ai — 1/ f ° r a l l m £ Z. Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 51 Next, we use the definition of the Reidemeister rewriting function (3.3) to express the second and third types of relations in (3.17) in terms of the generators s a * a . : • T(akTaa7k) = snk n.snk+in s~k+1 • s~k _ (3.18) T(akRia7k) = snk„.s„k+i„ snk+2 „ s~k+2 s~k+1 s"k (3.19) Taking i = 1, j > 3 in (3.18) we get Thus, by induction on k, = *l,a,- (3-20) for j > 3 and for all /c € Z. Therefore, Z„ is generated by sak a 2 = a^aT^" 1" 1^ and s i > a £ = a^aT1, where keZ,3<£<n.To simplify notation let us rename the genera-tors; let pk := a\a2a^k+r) and qe agaT1, for k 6 Z , 3 < £ < n. We now investigate the relations in (3.18) and (3.19). The relations in (3.19) break up into the following three types (using 3.20): Pk+iPkUPk1 (taking i = 1) (3.21) PkQ3Pk+2q3 Vfc-jW 1 (taking i = 2) (3.22) g i Q i + i ^ ^ W ^ i + i for 3 < i < n - 1. (3.23) The relations in (3.18) break up into the following two types: PkQjPkliQj1 for4< j <n, (taking i = 2) . (3.24) QiWi1 Q.J1 for 3 < i < j < n, \i - j\ > 2. (3.25) We now have a presentation for Zn consisting of the generators Pk,qe, where k G Z, 3 < t < n — 1, and defining relations (3.21) -(3.25). However, notice that relation (3.21) splits up into the two relations Pk+2 = Pk Vfe+i for k>0, (3.26) Pk= Pk+iPkU f o r k '< °- (3-27) Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 52 Thus, for k ^ 0,1, pk can be expressed in terms of po and p\. It follows that Zn is finitely generated. In order to show Zn is finitely presented we need to be able to replace the infinitly many relations in (3.22) and (3.24) with finitely many relations. This can be done using lemmas 3.3 and 3.4, but this requires us to add a new letter b to the generating set with a new relation b — po<?3£>o Thus Zn is generated by po, pi,qi, b, where 3 < £ < n— 1, with defining relations: P0Q3P01 =•&, PobPo1 = b2q7lb, piqzpT1 = q^b, P\bp7l = (q^b)3q^2b, PoQj = QjPi (4 < j < n), piq-j = qjP^pi (4 < j < n). QiQjQ^Qj1 (3 < i < j < n, | i - j | > 2). Noticing that for n = 2 the generators qk (3 < k < n), and b do not exist, and for n = 3 the generators qk (4 < k < n) do not exist, we have proved the following theorem. Theorem 3.6 For every n > 2 the commutator subgroup A'A of the Artin group AA„ is a finitely presented group. A'A is a free group with two free generators po = a2a71, pi = aia2a72-A'AS is the group generated by Po — a2a71-, pi = axa2a72, q = a3aT1, b = a2axla3a7l, with defining relations b = PoqPo1, PobPo1 = b2q'1b, PiQPT1 = q^b, PibpT1 = (q~1bfq-2b. For n> A the group A'AN is generated by p0 = a2a1~1, pi = aia2ar2, q3 = a3aT1, b = a2axla^a^1, qe — a^aV1 (4 < I < n — 1), Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 53 with defining relations b = p0q3Pol, Pobpo 1 = b2q^lb, Poqi = qiPl (4 < i < n), pxqi = qip0~1pi (4 < i < ri) Q3Qi = qm (5 < i < n), g3<?4<?3 = 9403Q4, QiQj = QjQi (4 < t < j - 1 < n - 1), qiqi+im = qi+iqiqi+1 (4 < % < n - 1). • Corollary 3.7 For n > 4 the commutator subgroup A'An of the Artin group of type An is finitely generated and perfect (i.e. A"An — A'A ). Proof. Abelianizing the presentation of A'A in the theorem results in a pre-sentation of the trivial group. Hence A"A — A'A . • Now we study in greater detail the group A'A3, the results of which will be used in section 4.2.1. From the presentation of A'A given in theorem 3.6 one can easily deduce the relations: Po 1QPO = qh~lq2: PolbPo = q, • Pilqpi = qb^qK pTlbPi = qb~lqA-Let T be the subgroup of A!Az generated by q and b. The above relations and the defining relations in the presentation for A'Aa tell us that T is a normal subgroup of A'A . To obtain a representation of the factor group A'Aa/T it is sufficient to add to the defining relations in the presentation for A'A the relations q = 1 and b = 1. It is easy to see this results in the presentation of the free group generated by po and pi. Thus,-A'A JT is a free group of rank,2, F2. We have the exact sequence 1 ^ T ^ A ' A 3 ^ A ' A J T - ^ 1 . • Since A'As JT is free then the exact sequence is actually split so where the action of F2 on T. is defined by the defining relations in the presen-tation of A'A and the relations above. In [GL69] it is shown (theorem 2.6) the group T is also free of rank 2, so we have the following theorem. Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 54 Theorem 3.8 The commutator subgroup A'As of the Artin group of type A% is the semidirect product of two free groups each of rank 2; A'M~F2*F2. • 3.3.3 TypeS The presentation of AB„ is Asn•= (ai, ...,an : a^j - c^a* for \i - j\ > 2, ajaj+iaj = a j + i a i Q i + i for 1 < i < n — 2 Let Tij, Ri (1 < i < n—2), and Rn-i denote the associated relators ciiaja^aj1, aiai+iaia^:1a71a^1, and an-ianan-iana'^a^a^a'1, respectively. As seen in figure 3.1 the graph (Bn)odd has two components: Ti and F2, where F2 denotes the component containing the single vertiex an. Let degx and deg 2 denote the associated degree maps, respectively, so from theorem 3.2 A'Bn = Zfn = {Ue ABn : degl(U) = 0 and deg2(U) = 0}. For simplicity of notation let Z$ be denoted by Zn. For elements U, V € AAU, znu = znv & uv-1 e zn . deg^U) = deg^V), and -deg2(U) = deg2(V), thus a Schreier system of right coset representatives for Asn modulo Zn is TZ = {akan : k,£e Z} By the Reidemeister-Schreier method, in particular equation (3.2), Zn is gen-erated by • = UWn^nia~{ \ 1 . if j = n. aia^aja^a-^ ^k+l^ if j ^ n Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 55 with presentation 2 n = (sakaen,a3- > • • • : Sap1aqn,ax> • • • > : T C a ^ T i j C o ^ ) - 1 ) , • • •, (1 < i < j < n, \i - j\ > 2), r l a J a ^ l a K ) - 1 ) , . . . , d < i < n - 2), r ( a ^ i ? n _ i ( a ^ ) - 1 ) , . . . } , (3.28) where p , q e Z , A e { l , . . . , n - l } such that a^anax ~ a^a&a.\ ("freely equal"). Again, our goal is to clean up this presentation. The cases n = 2, 3, and 4 are straightforward after one sees the computa-tion for the general case n > 5, so we will not include the computations for these cases. The results are included in theorem 3.9. From now on it will be assumed that n > 5. Since v a -p—q— fai + 1 an A ^ n n a^a9aA « a^a£aA = < -4=4> A = n or; A = 1 and g = 0, ya^ah X = n the first type of relations in (3.28) are precisely Sa\ai,an = 1> a n d S a k , a i = L ( 3 " 2 9 ) The second type of relations in (3.28), after rewriting using equation (3.3), are • Sakae„ AiSnkne a - a- S~T7 '—~ S~T~7 - l -1 '• . (3.30) where 1 < i < j < n, \i — j\ > 2. Taking i = 1 and 3 < j < n — 1 gives: for I = 0 (using (3.29)); Sak+\a} = Sak,a^ (331) so by induction on k, sak,a3 = s i ,a; f o r 3 < j < n - 1, (3.32) and for ^ # 0; Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 56 We will come back to relation (3.33) in a bit. Taking i = 1 and j = n in (3.30) (and using (3.29)) gives V a * a , S k e+i • ( 3- 34) a i " n > a i a j a „ + ,ai v ' So, by induction on £ (and (3.29)) we get s a k a i m = l forfe,£eZ. ' (3.35) Taking 2 < i < n - 2, i + 2 < j < n in (3.30) gives f o r j ( 3 3 6 ) S a K ^ S a f ^ , a i - • for j = n. In the case j — n induction on £ gives M ^ . a * = M . a i (2 < * ' < « - 2 ) . (3.37) So from (3.32) it follows {•5i a, 3 < i < n — 2 *1 • 7-2 ( 3 3 8 ) We come back to the case j < n — 1 later. Returning now to (3.33), we can use (3.35) to get s„t+i„£ „ ' = snkne „ (3 < j < n - 1). Thus, by induction on * a K . « i = s 4.a* (3 < J < " - ! ) • (3-39) For 3 < j < n — 2 we already know this (equation (3.38)), so the only new information we get from (3.33) is / Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 57 Collecting all the information we have obtained from T(a\anTij (a\an) l) 1 < i < j < n, \i — j\ > 2, we get: sakain,ai Sl,at 3<i<n-,2, .Sak,a2 i = 2, saka^l,an-1 sa£,an_i> and (from (3.36)), for 2 < i < n - 3 and i + 2 < j.<n-l, M < . ^ S a f + 1 a ^ , a / a J + 1 a £ , a i S a K . < » i ' This relation breaks up into the following cases (using (3.41)) (3.41) (3.42) S a* .a2 S l ' ° ^a* 1 + 1 , a2 S l i for i = 2, 4 < j < n - 2, - i snk n^sne n , s fc. i s , ' for i = 2, ?' = n — 1, • a1;a2 an,an_i Qfc+1)(l2 a*,a„_i J , si^si^jSri^Mj for 3 < i < n - 3, i + 2 < j < n - 2, I «i a,-sn« n i sVl s~e for 3 < i < n — 3, j = n — 1, (3.43) The third type of relations in (3.28); T(a\anRi(a\an) l), after rewriting using equation (3.3), are M a i , 0 l \ { + V « , a j + 1 \ f + 2 a ' „ , a I S A W A ( I A J + 1 S „ W A « 1 „ J S A { « « , T T I + 1 which break down as follows (using (3.41)): (3.44) s„k+i „ s k + 2 s k (i = 1), Sa*,o2Sl,a3Sa^+2,a2Sl,a3SQ^+\a2Sl,a3 1 --1 a = 2), for 3 < i < n - 3, I si a „ , s i a „ „s~} s r i s - / , (i = n — 2), ^ l , " n - 2 an,an_;i l , « n - 2 <j*,an_l l , 1 n - 2 a£,a„_l' v " (3.45) The fourth type of relations in (3.28); T(ak^anRn-i(af{an) 1), after rewriting using equation (3.3), is sne n , s„i+1 n s C+2 s e+l ) an,an-! an ,o„_i a* + 2 ,a n _! a& + 1 ,a„_i (3.46) Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 58 where we have made extensive use of the relations (3.41). From (3.41) it follows that Zn is generated by sak -a2, s ^ a i , and saen a n _ 1 for k, £ € Z and 3 < % < n — 2. For simplicity of notation let these generators be denoted by pkl Qi/ a n d rt> respectively. Thus, we have shown that the following is a set of defining relations for Zn: Pkqj=QjPk+i (4 < j < n - 2, k € Z), Pkre = rePk+i {k, £eZ), q.i%=°i<li (3 < i < j < n- 2, \i - j\ > 2), Qir'e = reqi (3 < i < n - 3), Pk+iPkUPk1 (fc€Z), (3.47) Pkq3Pk+2Q31Pkl1q31 (ke Z), q%qi+iqi = qi+iqm+i (3 < i < n - 3), qn-2reqn-2 = reqn-2re (£ € Z), rere+1r7+2re+i ( ^ z ) . The first four relations are from (3.43), the next four are from (3.45), and the last one is from (3.46). The fifth relation tells us that for k ^ 0,1, pk can be expressed in terms of po and p\. Similarly the last relation tells us that for £ ^ 0,1, re can be expressed in terms of ro and r\. From this it follows that Zn is finitely generated. Using lemmas 3.3 and 3.4 to replace the first, second and sixth relations, assuming we have added a new generator b and relation b = poqsPo1, we arrive at the following theorem. Theorem 3.9 For every n > 3 the commutator subgroup A'Bn of the Artin group Asn is a finitely generated group. Presentations for A'Bn, n > 2 are as follow s: A'B2 is a free group on countably many generators: .[<4ai] ( * e Z \ { 0 , ± l } ) , [aka2,ai} (k 6 Z\{0}) , A'B is a free group on four generators: [a^ 1 ,^ 1 ] , [a 3 ,a 2 ] [a^ 1 ,a2 : ], [ai',a2\[aT1,a2~l], [aia 3, a2\[a71, a 2 1 ] . / Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 59 A'B is the group generated by P k = a\a2a-[k+1) = [akl,a2][a^,a2l\1 (k € Z) qe = a{az{aia{)~1 = [ai,^]^1,^1]^1,^1], (t € Z), with defining relations s PkqePk+2 = qePk+iqe (k,£eZ), qeqe+i = qe+iqe+2 {3 <i <n-3). For n>5the group A!Bn is generated by , p0 = a2a1~1,' pi = a 1 a 2 a 1 ' 2 , g 3 = c^aT/1, re = anan-i(aian)~l (I £ Z), 6 = a 2 a ^ 1 a 3 a 2 X , = a ^ a ^ 1 (4 < z < n — 2), i f z'r/j defining relations Poqj = qjPi, Piqj = qjPoXPi C4 < J < n - 2), Pore = repi, pire = rep^Pi (£€%),. QiQj = qjqi C3 < i < j < n - 2, \i = j\ > 2), q%ri = nqt (3<i<n-3), Poq^Po1 = b , PobPo1 = b2q^lb, PiqzPi1 = q^b, ' Pibp\~l = (q^bfq^b, 'QiQi+iQi = qi+iqiQi+i (3 <i <n-3), qn-2Tiqn-2 = require (£ € Z), ren+ir^r^ (£ e Z), • Corollary 3.10 For n>5the commutator subgroup A'Bn of the Artin group of type Bn is finitely generated and perfect. Proof. Abelianizing the presentation of A'B in the theorem results in a pre-sentation of the trivial group. Hence A"Bn = A'B. ' • Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 60 3.3.4 Type£> The presentation of AD„ is Aon = ( a i , —,an : CLidj = ajdi for 1 < i < j < n - l,\i - j\ > 2, - , ano-j — ajan for j ^ n — 2, a j a j + i a j = a j + i a i a i + i for 1 < i < n — 2 a-n-2Q>n&n-2 = anan-2an). j As seen in figure 3.1 the graph {Dn)odd is connected. So by theorem 3.1 A'Dri=ZDn = {UeADn:deg(U) = 0}. The computation of the presentation of A'Dn is similar to that of A'A , so we will not include it. Theorem 3.11 For every n > 4 the commutator subgroup A'Dn of the Artin group Aon is a finitely presented group. A'Da is the group generated by po = a2aT1, pi = aia2a72, q3 = a^aV1, 04 = a^aT1, b = a2aT1a3a21, c = a2aT1a,4a2 1 , with defining relations b = PoqzPo1, PobPo1 = b2q^b, PiQ3Pil = <lzlb, PibpV1 = (<?3 16)3q3"26, c = P094P0 X> Pocpo 1 = c ^ T ^ c , P i Q ^ r 1 = Q4lc' PicPi1 = (04 1 c ) 3 o 4 " 2 c , / 93<?4 '= 9403-For n>5the group A'Dn is generated by . . . po = a2a7l, p\ = aia2a72, qi = a^aT1 (3 < £ < n), b = a2aTla3a2l, Chapter 3. Commuta tor Subgroups of Finite-Type Artin Groups 61 with defining relations * = Po93Po1' PobPo1 = tfq^b, . PiqsPr1 = %lb, PibpT1 = {q^bfq^b, PoQj = QjPu PiQj = QjP^Pi (4 < j < -n), QiQi+iQi = qi+iqiQi+l (3 < i < n - 2), qtqj = qjqi (3 < i < j < n - 1, \i - j\ > 2), qnqj = qjqn ( j ^ n - 2 ) . • - Corollary 3.12 For n> 5 the commutator subgroup A'Dn of the Artin group of type Dn is finitely presented and perfect. • 3.3.5 TypeE The presentation of A E N , n = 6,7, or 8, is . A.EN = (o-i; an : CLidj = djdi for 1 < i < j < n — l,\i — j\ > 2, a;a n — anai for i ^ 3, aiOi+idi = di+ididi+i for 1 < i < n — 2 a3anaz = , d n a 3 d n ) . As seen.in figure 3.1 the graph (En)odd is connected. So by theorem 3.1 A'En = ZEn = {U£ AEn •: deg(U) = 0}. The computation of the presentation of A'En is similar to that of A'An. Theorem 3.13 For n = 6,7, or 8 the commutator subgroup A'E of the Artin group AE„ is a finitely presented group. A'En is the group generated by po — d2dT1, pi = a^a^" 2 , q£ = a^dV1 (3 < £ < n), b = d2al~1 d^a^1-, / Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 62 with defining relations b = P0Q3P01, . Pobpo1 = b2q^b, PKl3Pil = q^b, Pibp^1 = [q^bfq^b, Poqj - qjPi, Piqj = qjP^Pi (4 < j < n), QiQi+iqi = qi+iqiqi+i (3 < % < n - 2), mqj = qjqt (3 < % < j < n - l , \i - j\ > 2), q%qn = qnqi ( 4 < v < n - i ) . Corollary 3.14 For n = 6,7, or 8 the commutator subgroup A'EN of the Artin group of type En is finitely presented and perfect. . . • 3.3.6 TypeF The presentation of AFA is -A F „ = (ai,,02,03,04 : didj = ajdi for \i - j\ > 2, o i a 2 o i = 0,20.10,2, 0,20,30,20,3 = a 3 a 2 a 3 a 2 , 030403 = 040304). As seen in figure 3.1 the graph (En)odd has two components: T i and T 2 , where T i denotes the component containing the vertices 01,02, and r 2 the component containing the vertices 03,04. Let degx and deg2 denote the asso-ciated degree maps, respectively, so from theorem 3.2 A'F4 = 4? = iU G M • d e § i ( ^ ) = 0 and deg2(U) = 0}. By a computation similar to that of Bn we get the following. Theorem 3.15 The commutator subgroup A'Fa of the Artin group of type F4 is the group generated by ~ . P k = aka2a-{k+l) = [ak,a2}[a^,af] (fceZ), qe = o ^ a 3 a 7 ( m ) = [a{,oslfaj1,03 l] {I G Z), Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 63 with defining relations PkqePk+iqe+i = qePkqe+iPk+i ( M e Z): The first two types of relations in the above presentation tell us that for k ^ 0,1, j>fc can be expressed in terms of po and p i , and similarly for qg. Thus A ' F 4 is finitely generated. However, A'Fi is not perfect since abelianizing the above presentation gives A'F/AFi c± Z 4 . 3.3.7 TypeS The presentation of AH„, n = 3 or 4, is Aff„ = ( a i , a n : a jO j = ajCii for | i - j\ > 2, aia2aia2ai = a2aia2a\a2, ajaj_|-iai = ai+iajaj+i for 2 < i < n — 1). 'As seen in figure 3.1 the graph (Hn)olid is connected. So by theorem 3.1 4 = 2tf„ = e -4//n : deg(U) = 0}. The computation of the presentation of A'Hn is similar to that of A'An. Theorem 3.16 For n = 3 or 4 f/ze commutator subgroup A'Hn of the Artin group AHu is the group generated by pk = aka2a^{k+1) (JfeeZ), qe = aea~e ( 3 < £ < n ) , wn'f/z defining relations PkQj = QjPk+i (4 < j < n), qiqi+iqi = gj+i<Mm (3 <.i < n - l ) . • / Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 64 The second relation tells us that for k ^ 0,1,2,3, pk can be expressed in terms of po,Pi,P2, and p3. Thus, A'Hn is finitely generated. Abelianizing the above presentation results in the trivial group. Thus, we have the following. Corollary 3.17 For n = 3 or A the commutator subgroup A'Hn of the Artin group of type Hn is finitely generated and perfect. • 3.3.8 T y p e / The presentation of h{m), m > 5, is Ar2(m) = (ai,Q2 : {aia2)m = ( a 2 a i ) m ) . In figure 3.1 the graph {h{fn))0dd is.connected for m odd and disconnected for m even. Thus, different computations must be done for these two cases. We have the following. Theorem 3.18 The commutator subgroup A'l2^ of the Artin group of type him), rh > 5, is the free group generated by the {m — l)-generators aka2a~ik+1) (k G {0,1,2,... ,m — 2}), when m is odd, and is the free group with countably many generators [ai,ai] {£eZ\{-{m/2-l)}), [a{ai,ai} ( f 6 Z , j = l ,2 m / 2 - 3 ) , {a™/2-2ai,ai} {£EZ\{m/2-l}), [aJo2,oi] (k € Z). when m is even. 3.3.9 Summary of Results r Table 3.1 summarizes the results in this section. The question marks (?) in the table indicate that it is unknown whether the commutator subgroup is finitely presented. However, we do know that for these cases the group is finitely generated. If one finds more general relation equivalences along the lines of lemmas 3.3 and 3.4 then we may be able to show that these groups are indeed finitely presented. Chapter 3. Commutator Subgroups of Finite-Type Artin Groups 65 TypeT finitely generated/presented perfect A  R -tin yes/yes n = 1, 2,3 : no, n > 4 : yes n = 2 : no, n > 3 : yes n = 2,3,4: no, B N , / n = 3 : yes, n > 3 : ? n > 5 : yes D N yes/yes n — 4 : no, n > 5 : yes EN yes/yes yes FA yes/? no H N yes/? yes I2 (m) (m even) no/no no (m odd) yes/yes no Table 3.1: Properties of the commutator subgroups Chapter 4 Local Indicability of Finite-Type Artin Groups Locally indicable groups first appeared in Higman's thesis [Hig40a] on group rings. He showed that if G is a locally indicable group and R an integral domain then the group ring RG has no zero divisors, no idempotents other than 0 and 1, and no units other than those of the form ug (u a unit in R, g G G). Higman's results have subsequently been extended to larger classes of groups, for example right-orderable groups. Our primary interest in local indicability is its application to the theory of right-orderability which is the topic of chapter 5. 4.1 Definitions A group G is indicable if there exists a nontrivial homomorphism G —> Z (called an indexing function). A group G is locally indicable if every nontriv-ial, finitely generated subgroup is indicable. Notice, finite groups cannot be indicable, so locally indicable groups must be torsion-free. Every free group is locally indicable. Indeed, it is well known that ev-ery subgroup of a free group is itself free, and since free groups are clearly indicable the result follows. Local indicability is clearly inherited by subgroups. The following simple theorem shows that the category of locally indicable groups is preseved under extensions. Theorem 4.1 IfK, H and G are groups such that K and H are locally indicable and 66 Chapter 4. Local Indicability of Finite-Type Artin Groups 67 fit into a short exact sequence 1 — • K —^—» G —^—> H — • 1, then G is locally indicable. Proof. Let g\,..., gn € G, and let (gi,.. • ,gn) denote the subgroup of G which they generate. If tp{{gi, •. •, gn)) i1 {1} then by the local indicability of H there exists a nontrivial homomorphism / : ip((gi,..., gn)) —• Z. Thus, the map foip:(g1,...,gn) —>Z is nontrivial. Else, if ip((gi,...,gn)) = {1} then gi, • • • ,gn £ kerV> = Im^ (by exactness), so there exist k\,..., kn € K such that <f>(ki) = gi, for all i. Since 4> is one-to-one (short exact sequence) then <f> : (k\,..., kn) —> (gi,..., gn) is an isomorphism. By the local indicability of K there exists a nontrivial homomorphism h : (k\,.,., kn) —• Z, therefore the map hod'1 : (gi,...,gn) —>Z is nontrivial. • Corollary 4.2 If G and H are locally indicable then so is G .© H. Proof. The sequence 1 — > H — ^ — > G ® H ^ ^ G — > 1 where 4>(h) = (1, h) and ip(g, h) = g is exact, so the theorem applies. • If G and H are groups and <j> : G —> Aut(iY). The semidirect product of G and # is defined to be the set H x G with binary operation (hi,gi)-(h2,g2) = {h\-g\*h2,gi92)' where g * h denotes the action of G on H determined by <fi, i.e. g * h := 4>(g)(h) £ H. This group is denoted by H G. Corollary 4.3 If G and H are locally indicable then so is H G. Chapter 4. Local Indicability of Finite-Type Artin Groups 68 g then kei-0 = H and 4 1 • The following theorem of Brodskii [Bro80], [Bro84], which was discovered independently by Howie [How82], [HowOO], tells us that the class of torsion-free 1-relator groups lies inside the class of locally indicable groups. Also, for 1-relator groups: locally indicable torsion free. Theorem 4.4 A torsion-free 1-relator group is locally indicable. To show a group is not locally indicable we need to show there exists a finitely generated subgroup in which the only homomorphism into Z is the trivial homomorphism. Theorem 4.5 IfG contains a finitely generated perfect sugroup then G is not locally indicable. Proof. The image of a commutator [a, b] := a6a _ 1 6 _ 1 under a homomor-phism into Z is 0, thus the image of a perfect group is trivial. • 4.2 The Local Indicability of Finite-Type Artin Groups Since finite-type Artin groups are torsion-free (see section 2.8), theorem 4.4 implies that the Artin groups of type A2, B2, and h{m) (m > 5) are locally indicable. In this section we determine the local indicability of all1 irreducible finite-type Artin groups. It is of interest to note that the discussion in section 3.2, in particular the-orem 3.2, shows that an Artin group Ar and its commutator subgroup A'v fit into a short exact sequence: 1 A'T — ^ ^ r — Z m —> 1, ' 1 with the exception of type F± which at this time remains undetermined. Proof. If ifi : H xi^ G —> G denotes the map (h,g) the groups fit into the exact sequence l _ > # _ ™ i L > H^G G Chapter 4. Local Indicability of Finite-Type Artin Groups 69 where m is the number of connected components in Foa\d, and cf> can be iden-tified with the abelianization map. Thus, the local indicability of an Artin group .Ar is completely determined by the local indicability of its commuta-tor subgroup A'R (by theorem 4.1). In other words, A r is locally indicable A'R is locally indicable. This gives another proof that the Artin groups of type A2, B2, and I2(m) (m > 5) are locally indicable, since their corresponding commutator subgroups are free groups as shown in Chapter 3. 4.2.1 Type A AA-L is clearly locally indicable since AAL — Z, and, as noted above, AA2 is also locally indicable. For AA3, theorem 3.8 tells us A'AA is the semidirect product of two free groups, thus A'A is locally indicable. It follows from our remarks above that AA3 is also locally indicable. As for AA„, n > 4, corollary 3.7 and theorem 4.5 imply that AAU is not locally indicable. Thus, we have the following theorem. Theorem 4.6 AA„ is locally indicable if and only ifn = 1,2, or 3. 4.2.2 T y p e B We saw above AB2 is locally indicable. For n = 3 and 4 we argue as follows. Let V^+l denote the (n + l)-pure braids in 2$ n + i = Ayi n , that is the braids which only permute the first n-strings. Letting bi, ...,bn denote the genera-tors of ABH a theorem of Crisp [Cri99] states Theorem 4.7 The map ABN — • AAU defined by bi i > dj, bn i > Qjn is an injective homomorphism onto V^+l. That is, ABN — VnXi '< ®n+i = AAU-Chapter 4. Local indicability of Finite-Type Artin Groups 70 By "forgetting the n t h-strand" we get a homomorphism / : V^Xi — > ®n which fits into the short exact sequence 1 — # — B n — 1, ' where K = ker f = {(3 <E V^Xl '• m e n r s t 71 strings °f P a r e trivial}. It is known that K ~ Fn, the free group of rank n. Since Fn is locally indicable and !B n (n = 3,4) is locally indicable then so is A B „ , for n = 3,4. Futhermore, the above exact sequence is actually a split exact sequence so AB„ — T^n+l ~ F, x <B„. " 1 • As for ^4jgn, n > 5, corollary 3.10 and theorem 4.5 imply that ^1B„ is not locally indicable, for n > 5. Thus, we have the following theorem. Theorem 4.8 Asn is locally indicable if and only ifn = 2,3, or 4. 4.2.3 TypeD It follows corollary 3.12 and 4.5 that Aon is not locally indicable for n > 5. As for ADA, we will show it is locally indicable as follows. A theorem of Crisp and Paris [CP02] says: Theorem 4.9 Let Fn-i denote the free group of rank n — 1. There is an action P • -4ji n_i —> A u t ( F n _ i ) such that Apn — Fn-\ xi AAu_! and p is faithful. Since AA3 and F3 are locally indicable, then so is AD A- Thus, we have the following theorem. Theorem 4.10 Aryn is locally indicable if and only ifn = 4.. 4.2.4 TypeS Since the commutator subgroups of[AE„, n = 6, 7,8, are finitely generated and perfect (corollary 3.14) then AEU is not locally indicable. 4.2.5 Type 7 Unfortunately, we have yet to determine the local indicability of the Artin group A F A -Chapter 4. Local Indicability of Finite-Type Artin Groups 71 4.2.6 T y p e S Since the commutator subgroups of Ann, n = 3,4, are finitely generated and perfect (corollary 3.17) then Ann is not locally indicable. 4.2.7 T y p e / As noted above, since the commutator subgroup A'l2^ of Ai2(m) (m > 5) is a free group (theorem 3.18) then A'l2^ is locally indicable and therefore so is •Ai2(m)• One could also apply theorem 4.4 to conclude the same result. Chapter 5 Open Questions: Orderabiiity In this chapter we discuss the connection between the theory of orderable groups and the theory of locally indicable groups. Then we discuss the current state of the orderabiiity of the irreducible finite-type Artin groups. 5.1 Orderable Groups A group or monoid G is right-orderable if there exists a strict linear ordering < of its elements which is right-invariant: g < h implies gk < hkior all g, h, k in G. If there is an ordering of G which is invariant under multiplication on both sides, we say that G is orderable or for emphasis bi-orderable . Theorem 5.1 G is right-orderable if and only if there exists a subset P c G such that:. V -V CV (subsemigroup), G\{1} = VUV-\ Proof. Given V define < by: g < h iff hg''1 G V. Given < take V = {g <E G : Kg}. • In addition, the ordering is a bi-ordering if and only if also gVg-yczV, Mg € G. The set V c G in the previous theorem is called the positive cone with respect to the ordering <. 72 Chapters. Open Questions: Orderability 73 The class of right-orderable groups is closed under: subgroups, direct products, free products, semidirect products, and extension. The class of or-derable groups is closed under: subgroups, direct products, free products, but not necessarily extensions. Both left-orderability and bi-orderability are local properties: a group has the property if and only if every finitely-generated subgroup has it. , Knowing a group is right-orderable or bi-orderable provides useful infor-mation about the internal structure of the group. For example, if G is right-orderable then it must be torsion-free: for 1 < g implies g < g2- < g3 < •••<<?"<•••. Moreover, if G is bi-orderable then G has no generalised torsion (products of conjugates of a nontrivial element being trivial), G has unique roots: gn = hn g = h, and if [gn, h] — 1 in G then [g, h] = 1. Further con-sequences of orderablility are as follows. For any group G, let ZG denote the group ring of formal linear combinations n\g\ + • • • nkgk-Theorem 5.2 IfG is right-orderable, then ZG has no zero divisors. Theorem 5.3 (Malcev, Neumann) IfG is bi-orderable, then ZG embeds in a division ring. Theorem 5.4 (LaGrange, Rhemtulla) If G is right order able and H is any grooup, then ZG ~ ZH implies G ~H It may be of interest of note that theorem 5.2 has been conjectured to hold for a more general class of groups: the class of torsion-free groups. This is known as the Zero Divisor Conjecture. At this time the Zero Divisor Conjecture is still an open question. The theory of orderable groups is well over a hundred years old. For a general exposition on the theory of orderable groups see [MR77] or [KK74]. Conrad [Con59] investigated the structure of arbitrary right-orderable groups, and defined a useful concept which lies between right-invariance and bi-invariance. A right-ordered group (G, <) is said to be of Conrad type if for all a, b e G, with 1 < a, 1 < b there exists a positive integer TV such that a < aNb. The following theorems gives the connection between orderable groups and locally indicable groups (see [RR02]). Theorem 5.5 For a group G we have Chapter 5. Open Questions: Orderabiiity 74 bi-ordemble => locally indicable =>• right-orderable. Theorem 5.6 A group is locally indicable if and only if it admits a right-ordering of Conrad type. One final connection between local indicability and right-orderability was given by Rhemtulla and Rolf sen [RR02]. Theorem 5.7 (Rhemtulla, Rolfsen) Suppose (G, <) is right-ordered and there is a finite-index subgroup H of G such that (H,<) is a bi-ordered group. Then G is locally indicable. An application of this theorem is as follows. It is known that the braid groups Q3n = i ^ n _ , are right orderable [DDRW02] and that the pure braids Vn are bi-orderable [KR02]. However, theorem 4.6 tells us that 23„ is not lo-cally indicable for n > 5 therefore, by theorem 5.7, the bi-ordering on Vn and the right-ordering on 23„ are incompatible for n > 5. - 5.2 Finite-Type Ar t in Groups The first proof the that braid groups 23n enjoy'a right-invariant, total ordering was given in [Deh92], [Deh94]. Since then several quite different approaches have been applied to understand this phenomenon.1 However, it is unknown whether all the irreducible finite-type Artin groups are right-orderable. For a few cases one can use theorem 5.6 along with the results of chapter 4 to conclude, right-orderability. One approach is to reduce the problem to showing that the positive Artin monoid is right-orderable. 5.2.1 Ordering the Monoid is Sufficient We will show that for a Coxeter graph F of finite-type the Artin group Ar is right-orderable (resp. bi-orderable) if and only if the Artin monoid Ar is right-orderable (resp. bi-orderable). One direction is of course trivial. 1For a wonderful look at this problem and all the differents approaches used to un-derstand it see the book [DDRW02]. Chapter 5. Open Questions: Orderabiiity 75 Let Ar be an Artin group, of finite-type. Recall that theorems 2.17 and 2.18 tell us that: For each U € Ar there exist Ui,U2 € At, where U2 is central in Ar such that dering. We wish to prove that Ar is right-orderable. The following lemma indicates how we should extend the ordering on the monoid to the entire group. Lemma 5.8 IJU& Ar has two decompositions; ; u = I W 2 _ 1 = UjJ?, where Ui,Ui <E Ap and U2, U2 central in Ar, then Ux <+ y2*=*Ui <+U2. Proof. U = UiU2~l' = U\U2 1 implies U\U2=PU\U2, since U2, U2 central and 4^p" canonically injects in Ar-If (7i <+ U2 then where the last implication follows from the fact that ii'Ui + > U2 then either: (i) Ui =- U2, in which case [7 = 1 and so Ui = U2. Contradiction, (ii) Ui + > U2, in which case UiU2 + > U2U2. Again, a contradiction. =» UiU2 <+ U2U2 => UiU2 <+ U2U2 => Z/1C/2 < + since < + right-invariant, since U2 central, since UiU2 =p U\U2, The reverse implication follows by symmetry. • This lemma shows that the following set is well defined: V = {U e Ar : U has decomposition U = UiU2~l where U2 <+ U\). Chapter 5. Open Questions: Orderability 76 It is an easy exercise to check that V is a positive cone in Ar which con-tains V+: the positive cone in A^ with respect to the order < + . Thus, the right-invariant order < + on A^ extends to a right-invariant order < on Ar-Furthermore, one can check that if < + is a bi-invariant order on A^ then V satisfies: UVU^cV, VUeAr-Thus, the bi-invariant order < + on A^ extends to a bi-invariant order < on Ar. Open question. Determine the orderability of the finite-type Artin monoids by giving an explicit order condition. 5.2.2 Reduction to Type E8 Table 2.1 shows that every irreducible finite-type Artin group injects into one type A, D, or E. 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Index a-chain, see chain,a-chain abelianize, 42 Artin extra large type, 36 finite-type, 19 group, 19 degree, 42 pure Artin group, 19 irreducible, 19, 34 monoid, 20 length, 20 parabolic subgroup, 36 system, 18 bilinear form B, 7 chain a-chain, 22 source, 22 target, 22 elementary, 22 source, 22' target, 22 imminent, 24 length, 23 primitive, 22 source, 22 target, 22 simple, 24 chain operator, 24 common divisor, 28 common multiple, 28 commutator, 41 commutator subgroup, 41 conjugacy problem, 14 Coxeter element maximal length, 10 extra large type, 15 graph, 5 finite-type, 15 positive definite, 15 type, 15 group, 5 universal, 5 length, 6 matrix, 4 finite-type, 15 parabolic subgroup, 11 system, 5 rank, 5 deletion condition, 11 divides, 21 on the left, 21 elementary simplification first kind, 14 second kind, 14 elementary transformation, 20 exchange condition, 10 strong, 10 extension, 66 freely equal, 41 82 fundamental element, 34 quadratic factor, 32 greatest common divisor, 28 group Artin, see Artin, group Coxeter, see Coxeter, group perfect, 44 symmetric, 5 group ring, 73 hyperplane arrangement of, 38 central, 38 simplicial, 38 indicable, 66 irreducible, 13 least common multiple, 28 length, 19 linear, 36 locally indicable, 1,66 matrix Artin, see Artin, matrix Coxeter, see Coxeter, matrix orderable, 1,72 bi-orderable, 1,72 Conrad type, 73 right, 1,72 positive cone, 72 positive definite, 7 positive equivalent, 20 positive semidefiriite, 8 positive transformation, 20 reduced expression, 6 reduction property, 20,21 reflection, 7,38 Reidemeister rewriting function, 41 Reidemeistef-Schreier method, 40 reverse, 21 root, 9 negative, 9 positive, 9 simple, 9 root system, 8 Schreier-system, 40 semidirect product, 67 sequence a-sequence, 29 square-free, 32 subgraph induced, 13 system Artin, see Artin, system Coxeter, see Coxeter, system word problem, 14 Zero Divisor Conjecture, 73 83 

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