September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

International Journal of Algebra and Computation c World Scientific Publishing Company °

ORDERINGS ON ARTIN-TITS GROUPS

HERVE SIBERT∗ Laboratoire de Math´ ematiques Nicolas Oresme, Universit´ e de Caen, BP5186 14032 Caen, France [email protected]

We prove that a construction similar to that described by Dehornoy in the case of braids is possible for every Artin-Tits group, yielding a partial ordering. A necessary condition for this partial order to be linear is that the associated Coxeter graph consists only of disjoint lines. So, in particular, type D is dismissed. Keywords: braid groups; Artin-Tits groups; acyclicity; orderings

1. Introduction In 1992, Dehornoy proved that Artin braid groups are provided with a left–order (i.e., an order compatible with the product of the left), based on the expression of braids in terms of the Artin generators σi . The braid order is defined as follows: we say that x < y is true when, amongst all the expressions of x−1 y, there exists at least one that contains a generator σi , but neither σi−1 , nor σj±1 with j < i. The fact that the relation < thus defined is a total order on Bn relies on two main properties, namely the Acyclicity Property, which claims that a braid admitting at least one expression where some σi appears but σi−1 does not is never trivial, and the Comparison Property, which claims that every braid has at least one expression where σ1 or σ1−1 does not appear. There exist now several proofs of these two properties, relying on very different approaches to the braid group: algebraic, combinatorial, topological, see [5]. Artin-Tits groups are a natural generalization of Artin braid groups. They appear explicitly in [8], and their structure was independently studied by Brieskorn and Saito in [1] on the one hand, and Deligne in [6] on the other hand. They are obtained by considering presentations resembling Artin’s presentation of the braid group, but involving relations of arbitrary length. It is natural to ask to what extent the braid order can extend to general ArtinTits groups, and this is our purpose in this paper. To this end, we study the counterpart of the acyclicity and comparison properties in these groups. Artin-Tits groups are associated with Coxeter graphs, and we say that an Artin-Tits group is of linear ∗ Current

address : ST-NXP Wireless, 9 rue Maurice Trintignant, 72094 Le Mans Cedex 9, France. 1

September 10, 2008

2

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

type when the associated graph consists of one line. With this definition, our main results are as follows: Proposition A. The Acyclicity Property extends to all Artin-Tits groups. Proposition B. The Comparison Property extends only to some direct products of linear type Artin-Tits groups. In the first section, we introduce Artin-Tits groups via Coxeter graphs and matrices, and recall the action of a family of Artin-Tits groups, the small type ArtinTits groups, on the free group, as defined in [3]. In the second section, we prove that the acyclicity property extends to all Artin-Tits groups. We first prove the result for a restricted family of Artin-Tits groups. Then, we describe an operation called unfolding, due to Crisp and Paris. It consists in mapping an Artin-Tits group onto another Artin-Tits group with a more simple structure. This construction enables us to extend acyclicity to all Artin-Tits groups. In the third section, we construct partial left–orders on general Artin-Tits groups, and we review well-known cases for which these orders are total. In the last section, we prove that a necessary condition for one of these orders to be total is that the Coxeter graph consists of disjoint lines only. In this case, the question whether one of these orders is indeed total remains open when there is a relation of length more than 3 in the considered presentation. 2. Artin-Tits groups We remind that the braid group Bn —the group of braids with n-strands— is the group with generators σ1 , . . . , σn−1 , satisfying relations σi σj σi = σj σi σj for |i − j| = 1, and σi σj = σj σi for |i − j| ≥ 2. One can also write these relations prod[σi , σj ; mσ ,σ ] = prod[σj , σi ; mσ ,σ ], where prod[σi , σj ; mσ ,σ ] is the product i j j i i j σi σj σi · · · with mσ ,σ terms, setting mσ ,σ = 3 for |i − j| = 1, and mσ ,σ = 2 for i j i j i j |i − j| ≥ 2. Artin-Tits groups are defined analogously, but with an arbitrary choice of the coefficients mi,j in the set {2, 3, 4, . . . , ∞} satisfying mj,i = mi,j . By convention, mi,j = +∞ means that there is no relation between σi and σj . Thus, the starting point for defining an Artin-Tits group is the data consisting of a family (finite or infinite) of generators, whose set we will denote by S, and of the double sequence of the coefficients ms,t for s, t ∈ S. By convention, we set ms,s = 1. We associate with these data an unoriented graph, defined as follows: the set of vertices is S, and two elements s and t of S are connected by an edge for ms,t ≥ 3, this edge being labelled with the coefficient ms,t (= mt,s ). Definition 1. A Coxeter graph is an unoriented graph Γ whose edges are labelled with coefficients in the set {n ∈ N, n ≥ 3}∪{∞}. We denote by SΓ the set of vertices of Γ. With every Coxeter graph Γ, we associate a Coxeter matrix MΓ , defined by MΓ = (ms,t )s,t∈S , where the coefficients ms,t are defined by: Γ (i) ms,s = 1 for every s in SΓ , (ii) for every pair of distinct elements of SΓ , ms,t is equal to the label of edge (s, t)

September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Orderings on Artin-Tits groups

3

if it exists, otherwise ms,t is 2. For convenience, we call the label of an edge its weight. By convention, only weights equal to or more than 4 are mentioned in a Coxeter graph. If the weight of each edge is 3, we say that the Coxeter graph is of small type. If there is an edge of infinite weight, then the corresponding Artin-Tits group contains a copy of the free group on two generators, so we say that the Coxeter graph is of free type. If the weight of each edge is finite, we say that the Coxeter graph is of non-free type. Figures 1 and 3 are examples of Coxeter graphs of small type. The Coxeter graph of Figure 2 defines an Artin-Tits group with one relation of length 4, the other relations being of length 2 and 3.

Fig. 1. The Coxeter graph An .

Fig. 2. The Coxeter graph Bn .

Definition 2. Let Γ be a Coxeter graph, and MΓ = (ms,t )s,t∈S be the associated Γ Coxeter matrix. The Artin-Tits group AΓ associated with Γ, is defined to be the group with presentation hSΓ ; prod[s, t; ms,t ] = prod[t, s; mt,s ], ms,t 6= ∞, s, t ∈ SΓ i, where prod[s, t; ms,t ] denotes the product sts · · · with ms,t terms. When the Coxeter graph is of small (resp. free, non-free) type, we say that the Artin-Tits group is of small (resp. free, non-free) type. The presentation of AΓ also defines a monoid denoted by A+ Γ.

Fig. 3. The Coxeter graph Dn .

September 10, 2008

4

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

2.1. Action on the free group In order to extend the acyclicity property to Artin-Tits groups and to construct orders on them, we are going to use the representation of small type Artin-Tits groups as groups of automorphisms of the free group, as introduced in [3]. This is a faithful representation that stems from the fact that every small type Artin-Tits group embeds in a mapping class group. Definition 3. Let Γ be a small type Coxeter graph, and Ω be a total order on SΓ . We denote by FΓ,Ω the free group based on α ∪ β, with α = {αs ; s ∈ SΓ } and β = {βs,t ; s, t ∈ SΓ , s Ω c1 >Ω b1 , and b1 >Ω s for every other vertex s of Γ0 . We

September 10, 2008

24

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

Fig. 9. Construction of the unfolding Γ0 when the edge (b, c) has weight 3.

consider the action of AΓ0 on the free group FΓ0 ,Ω associated to the order Ω. Let w = αa2 βb−1 α , and compute the reduced word x ˜0 (w). 1 ,a2 b1 QN −1 The ai ’s commute, so we obtain i=1 a ˜i (αa2 ) = αa2 . For every i 6= 2, we have

September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Orderings on Artin-Tits groups

25

QN −1 −1 a2 >Ω b1 >Ω ai . This implies i=1 a ˜i (βb2 ,a1 ) = βb−1 . At last, for every i 6= 2, the 2 ,a1 Q vertices ai and b1 are not linked together, which yields i6=2 a ˜−1 i (αb1 ) = αb1 . Thus, we obtain N Y

−1 −1 a ˜−1 ˜−1 2 (αb1 ) = βb1 ,a2 αb1 = w2 . i (w) = αa2 βb1 ,a2 a

i=1

For every i 6= 1, the inequality a2 >Ω b1 >Ω ci holds. We deduce N Y

−1 −1 −1 −1 −1 −1 c˜−1 ˜−1 1 (βb1 ,a2 ) = βc1 ,a2 αc1 βc1 ,a2 βb1 ,a2 βb1 ,c1 αc1 βb1 ,c1 . i (βb1 ,a2 ) = c

i=1

Similarly, we have

QN

˜−1 i (αb1 ) i=1 c

N Y

= βb1 ,c1 αc−1 βb−1 α , which gives 1 1 ,c1 b1

−1 −1 −1 c˜−1 i (w2 ) = βc1 ,a2 αc1 βc1 ,a2 βb1 ,a2 αb1 = w3 .

i=1

The bi ’s act trivially on all the letters of this reduced word, apart from αc−1 . There1 QN −1 −1 α β β α = w . We eventually obtain fore, we have i=1 ˜bi (w3 ) = βc−1 4 c1 ,a2 b1 ,a2 b1 1 ,a2 c1 N Y

a ˜i (w4 ) = βc−1 a ˜ (αc−1 )βc1 ,a2 βb−1 a ˜2 αb1 1 ,a2 2 1 1 ,a2

i=1

= βc−1 α−1 βc1 ,a2 αa−1 βc−1 β β −1 β α β −1 α 1 ,a2 c1 2 1 ,a2 c1 ,a2 b1 ,a2 b1 ,a2 a2 b1 ,a2 b1 = βc−1 α−1 βc1 ,a2 βb−1 α , 1 ,a2 c1 1 ,a2 b1 so the automorphism x ˜0 does not map P (αa2 ) into itself. As a2 is the maximal element in order Ω, we deduce from Proposition 11 that the element x0 is neither a2 -positive, nor a2 -neutral. Now, let Ω0 be the order obtained from Ω by exchanging b1 and c1 in Ω, and by replacing b1 by c1 in the word w. One shows, by considering the action of AΓ0 −1 associated with Ω0 on the free group FΓ0 ,Ω0 , that the element x0 is not a2 -positive, 0 nor a2 -neutral, so x is not a2 -negative. Therefore, x0 is not a2 -definite in AΓ0 and, by Lemma 39, the element x is not a-definite in the group AΓ . Lemma 43. If the vertices b and c are linked by an edge of weight at least 4, the element abc−1 a−1 is not a-definite. Proof. Let N = lcm{ms,t − 1, s, t ∈ S}, and denote by ai , bi and ci , with i = 1, . . . , N , the copies of vertices a, b and c in Γ0 . We consider the unfolding Γ0 defined by Figure 10. Let x = abc−1 a−1 , and x0 be the image of x by Φ in AΓ0 . Let Ω be an order on the vertices of Γ0 satisfying a2 >Ω c1 >Ω b1 >Ω b3 , and b3 >Ω s for every other vertex s of Γ0 . We consider the action of AΓ0 associated with the order Ω on the free group FΓ0 ,Ω . Let w = αa2 βb−1 α , and compute the reduced word x ˜0 (w). 1 ,a2 b1

September 10, 2008

26

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

Fig. 10. Construction of the unfolding Γ0 when the edge (b, c) has a weight at least 4.

QN −1 The ai ’s commute together, so we have i=1 a ˜i (αa2 ) = αa2 . The inequalities QN −1 −1 a2 >Ω b1 >Ω ai for i 6= 2 imply i=1 a ˜i (βb2 ,a1 ) = βb−1 . At last, for every i 6= 2, 2 ,a1

September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Orderings on Artin-Tits groups

27

the vertices ai and b1 are not linked. This gives a ˜−1 i (αb1 ) = αb1 . Thus, we obtain N Y

−1 −1 a ˜−1 ˜−1 2 (αb1 ) = βb1 ,a2 αb1 = w2 . i (w) = αa2 βb1 ,a2 a

i=1

For every i 6= 1, we have the inequalities a2 >Ω b1 >Ω ci . We deduce N Y

−1 −1 −1 −1 −1 −1 c˜−1 ˜−1 1 (βb1 ,a2 ) = βc1 ,a2 αc1 βc1 ,a2 βb1 ,a2 βb1 ,c1 αc1 βb1 ,c1 . i (βb1 ,a2 ) = c

i=1

Similarly, we have

QN

˜−1 i (αb1 ) i=1 c

N Y

= βb1 ,c1 αc−1 βb−1 α , which implies 1 1 ,c1 b1

−1 −1 −1 c˜−1 i (w2 ) = βc1 ,a2 αc1 βc1 ,a2 βb1 ,a2 αb1 = w3 .

i=1

Notice that this calculation is the same as in Lemma 42, because we have constructed Γ0 such, that the vertex b1 is linked only with c1 among the vertices ci ’s. The bi ’s act trivially on every letter of this reduced word, apart from αc−1 . This 1 comes from the choice of Ω, in which a2 and c1 are consecutives. On the other hand, as opposed to the case of Lemma 42, two copies of b act on αc−1 : these are b1 and 1 b3 . Then, we find N Y

−1 −1 −1 ˜bi (w3 ) = β −1 β −1 α β c1 ,a2 b1 ,c1 b1 b1 ,c1 βb3 ,c1 αb3 βb3 ,c1 αc1 βc1 ,a2 βb1 ,a2 αb1 = w4 .

i=1

QN The action of i=1 ai on w4 depends on the weight p of edge (a, b). For p = 3, vertices a2 and b3 are not linked, so only a4 acts on αb3 . For p = 4, there is no edge between vertices b3 and a4 , and only a2 acts on αb3 . At last, for the case p > 4, both generators a2 and a4 act on αb3 . However, notice that, as in previous cases, by the choice of Ω, the ai ’s do not act on one of the β’s in w4 . Thus, we obtain, for every p, N Y

a ˜i (w4 ) = βc−1 β −1 a ˜ (αb1 )βb1 ,c1 βb−1 a ˜4 a ˜2 (αb3 )βb3 ,c1 a ˜2 (αc−1 )βc1 ,a2 βb−1 α 1 ,a2 b1 ,c1 2 1 3 ,c1 1 ,a2 b1

i=1

= βc−1 β −1 β α β −1 α β β −1 1 ,a2 b1 ,c1 b1 ,a2 a2 b1 ,a2 b1 b1 ,c1 b3 ,c1 a ˜4 a ˜2 (αb3 )βb3 ,c1 αc−1 βc1 ,a2 αa−1 βc−1 β β −1 α 1 2 1 ,a2 c1 ,a2 b1 ,a2 b1 = βc−1 β −1 β α β −1 α β β −1 1 ,a2 b1 ,c1 b1 ,a2 a2 b1 ,a2 b1 b1 ,c1 b3 ,c1 a ˜4 a ˜2 (αb3 )βb3 ,c1 αc−1 βc1 ,a2 αa−1 βb−1 α 1 2 1 ,a2 b1

September 10, 2008

28

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

Owing to the value of p, the result is N Y

a ˜i (w4 ) = βc−1 β −1 β α β −1 α β β −1 1 ,a2 b1 ,c1 b1 ,a2 a2 b1 ,a2 b1 b1 ,c1 b3 ,c1

i=1

αb3 βa−1 α−1 βa4 ,b3 βb3 ,c1 αc−1 βc1 ,a2 αa−1 βb−1 α for p = 3, 1 2 4 ,b3 a4 1 ,a2 b1 N Y

a ˜i (w4 ) = βc−1 β −1 β α β −1 α β β −1 1 ,a2 b1 ,c1 b1 ,a2 a2 b1 ,a2 b1 b1 ,c1 b3 ,c1

i=1

βb3 ,a2 αa2 βb−1 βb−1 α β α−1 βc1 ,a2 αa−1 α for p = 4, and 2 3 ,a2 b3 b3 ,c1 c1 1 ,a2 b1 N Y

a ˜i (w4 ) = βc−1 β −1 β α β −1 α β β −1 β α 1 ,a2 b1 ,c1 b1 ,a2 a2 b1 ,a2 b1 b1 ,c1 b3 ,c1 b3 ,a2 a2

i=1

βb−1 α β −1 α−1 βa4 ,b3 βb3 ,c1 αc−1 βc1 ,a2 αa−1 βb−1 α for p > 4. 1 2 3 ,a2 b3 a4 ,b3 a4 1 ,a2 b1 In every case, we conclude that the automorphism x ˜0 does not map P (αa2 ) into itself. As a2 is the maximal element in order Ω, we deduce, by Proposition 11, that the element x0 is neither a2 -positive, nor a2 -neutral. −1 In order to show that x0 is also neither a2 -positive, nor a2 -neutral, we consider the graph obtained by exchanging the edges between the bi ’s and the cj ’s in Γ0 , i.e., by replacing every edge (bi , cj ) by an edge (bj , ci ). Then, this graph is, like Γ0 , an unfolding of Γ. Consider an order Ω0 satisfying a2 >Ω0 b1 >Ω0 c1 >Ω0 c3 , and c3 >Ω0 s for every other vertex s in the graph, and replace b1 by c1 in the word w. We −1 get the desired result – that x0 is neither a2 -positive, nor a2 -neutral —, similarly, by considering the action of AΓ0 associated with Ω0 on the free group FΓ0 ,Ω0 . Therefore, x0 is not a2 -definite in AΓ0 , and, by Lemma 39, the element x is not a-definite in the group AΓ . Proof of Lemma 40. Consider the element x of AΓ defined by x = abc−1 a−1 . In case, respectively, that the vertices b and c of Γ are not linked by an edge, are linked by an edge of weight 3, or are linked by an edge of weight at least 4, it follows respectively from Lemmas 41, 42 and 43 that the element x is not a-definite. As x is s-neutral for every s distinct from a, b, c, it is therefore an element not a-definite, and s-neutral for every vertex s of Γ other than a, b and c. Lemma 44. Every order on a non-free type Coxeter graph containing a cycle is non-extendible. Proof. Let Γ a non-free type Coxeter graph containing a cycle, and (s1 , . . . , sn ) a cycle of Γ. Then, by Lemma 40, for every i in {1, 2, . . . , n}, there exists an element xsi of AΓ not si -definite in AΓ , and s-neutral for every vertex s of Γ other than si and the two vertices adjacent to si in the cycle. Let then Ω be an arbitrary order on Γ, and let si be the maximal element among s1 , . . . , sn for Ω. The element xsi is not si -definite, and is s-neutral for every s >Ω si . b on A deduced from Ω is not total, so Ω is not extendible. Therefore, the order Ω Γ

September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Orderings on Artin-Tits groups

29

For the case when Γ contains a vertex of degree at least 3, we have: Lemma 45. Let Γ be a non-free type Coxeter graph, and let a be a vertex of degree at least 3. For every triple (b, c, d) of distinct vertices adjacent to a, there exists an element of AΓ that is not b-definite, and is s-neutral for every vertex s of Γ other than a, b, c and d. Proof. If vertex b is linked to c (respectively d) in Γ, then it is of degree at least 2, and, from the proof of Lemma 40, we deduce that the element x = abc−1 a−1 (respectively x = abd−1 a−1 ) is of the desired type. Now, suppose that b is neither linked to c, nor to d. Let N be the least common multiple of {ms,t − 1, s, t ∈ S}. Denote by ai , bi , ci and di , with i = 1 . . . N , the copies of vertices a, b, c and d in Γ0 . We consider an unfolding Γ0 such that vertices b1 , c1 and d1 are linked only to a2 among the ai , as described in Figure 11. Let x = badc−1 a−1 b−1 . We are going to prove that x is not b-definite, by considering its image x0 under Φ in AΓ0 . As b is neither linked to c, nor to d, the graph Γ0 does not contain edges linking the bi ’s to the cj ’s or to the dk ’s. The existence or not of an edge linking c and d does not interfere in the following calculation, so we do not represent such an edge in Figure 11. Let Ω be an order on Γ0 satisfying b1 >Ω c1 >Ω a2 >Ω d1 and d1 >Ω s for every other vertex s of Γ0 . Consider the action of AΓ0 on the free group FΓ0 ,Ω associated with Ω, and let w = αb1 βa−1 α β α−1 . We compute the reduced 2 ,b1 a2 a2 ,c1 c1 word x ˜(w). First, notice that the only nontrivial action of the bi ’s on w is that of b1 on αa2 . This gives N Y

−1 −1 −1 ˜b−1 (w) = α β −1 ˜b−1 (α )β b1 a2 ,b1 1 a2 a2 ,c1 αc1 = βa2 ,b1 αa2 βa2 ,c1 αc1 = w2 . i

i=1 −1 The action of Φ(a−1 ) on w2 reduces to that of a−1 2 on αc1 , and we obtain: N Y

−1 −1 −1 −1 a ˜−1 ˜−1 2 (αc1 ) = βa2 ,b1 βa2 ,c1 αc1 = w3 . i (w2 ) = βa2 ,b1 αa2 βa2 ,c1 a

i=1

Then, we observe that the only ci acting nontrivially on w3 is c1 , which, as we have b1 >Ω c1 >Ω a2 , acts on βa−1 . We get 2 ,b1 N Y

−1 −1 −1 c˜i−1 (w3 ) = c˜−1 1 (βa2 ,b1 )βa2 ,c1 αc1 = βa2 ,b1 βa2 ,c1 = w4 .

i=1

At last, as all the di ’s are smaller than a2 , b1 and c1 in the order Ω, we deQN ˜ −1 duce i=1 di (w4 ) = βa2 ,b1 βa2 ,c1 = w4 . We obtain successively the equalities QN QN ˜i (w4 ) = βa−1 β = w4 , and i=1 ˜bi (w4 ) = βa−1 β = w4 . i=1 a 2 ,b1 a2 ,c1 2 ,b1 a2 ,c1 Thus, the automorphism x ˜0 does not map P (αb1 ) into itself. As b1 is the greatest element for Ω, Proposition 11 implies that the element x0 is neither b1 -positive, nor b1 -neutral.

September 10, 2008

30

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

Fig. 11. Construction of Γ0 when Γ has a vertex of degree at least 3.

September 10, 2008

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Orderings on Artin-Tits groups

31

Now, let Ω0 and w0 be the order and the word obtained from Ω and w by exchanging c1 and d1 . Similarly, considering the action of AΓ0 associated with Ω0 on −1 the free group FΓ0 ,Ω0 , and computing the reduced word x˜0 (w0 ), we obtain that the −1 element x0 is neither b1 -positive, nor b1 -neutral. Therefore, x0 is not b1 -negative. Thus, x0 is not b1 -definite in A(Γ0 ), and, by Lemma 39, the element x is not b-definite in group AΓ . At last, x is s-neutral for every s distinct from a, b, c, d. Lemma 46. Every order on a non-free type Coxeter graph containing a vertex of degree at least 3 is non-extendible. Proof. Let Γ be a non-free type Coxeter graph containing a vertex a of degree at least 3, and denote by b, c and d three distinct vertices adjacent to a. By Lemma 45, for every t in {b, c, d}, there exists an element xt of AΓ not tdefinite in AΓ , and s-neutral for every vertex s distinct from a, b, c, d. Moreover, as the vertex a is of degree at least 2, we deduce from Lemma 40 the existence of an element xa not a-definite in AΓ , and s-neutral for every vertex s other than a, b, c, d. Now, let Ω be an arbitrary order on Γ, and let t be the maximal element in (a, b, c, d) for Ω. The element xt is s-neutral for every s >Ω t, and it is not b on A is not total, so Ω is not extendible. t-definite. Therefore, the deduced order Ω Γ From Lemmas 44 and 46, we deduce Proposition 31 in the non-free case. As a free type Coxeter graph has no extendible order (Lemma 33), the proof of Proposition 31 in the general case is now complete. As we emphasized at the beginning of the section, the question whether a product of non-free type linear Coxeter graphs admits an extendible order is open, apart for the small type case, where the complete result is given in Proposition 29. 6. Conclusion We have proved that, among the two properties that are instrumental in the construction of the canonical braid order, one—the acyclicity property—extends to general Artin-Tits groups, but, almost always, the other does not. This leads to the definition of left-orders which are strictly partial, apart from some particular cases, like direct products of braid groups. Therefore, the combinatorial approach to the braid order is only partly relevant for general Artin-Tits groups. This naturally leads to the (mostly open) question of whether the alternative approaches to the braid ordering [5] could prove more suitable. Let us mention that, in the case of the Artin-Tits group of type D4 , one can construct a total left-order using mapping class groups and triangulations along the lines of [5, chap.8] but, so far, no combinatorial characterization of the elements bigger than 1 in terms of the generators is known. If such a characterization was found, it could give clues for the definition of ”new” total left-orders on Artin-Tits groups.

September 10, 2008

32

14:10

WSPC/INSTRUCTION FILE

orderings˙artin

Herv´ e Sibert

Acknowledgements The author thanks Patrick Dehornoy for his help, comments and support during the preparation of this work, and Matthieu Picantin for sharing his wide knowledge of the subject. References [1] E. Brieskorn & K. Saito, Artin-Gruppen und Coxeter-Gruppen, Inventiones Math. 17 (1972) 245-271. [2] J. Crisp, Injective maps between Artin groups, in Geometric Group Theory Down Under, Proceedings of a Special Year in Geometric Group Theory, eds. J. Cossey et al, W. de Gruyter, Berlin (1999). [3] J. Crisp & L. Paris, Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, arXiv:math.GR/0212138, preprint (2002). [4] P. Dehornoy, Deux propri´et´es des groupes de tresses, C. R. Acad. Sci. Paris S´er.I 315 (1992) 633-638. [5] P. Dehornoy, I. Dynnikov, D. Rolfsen & B. Wiest, Why are braids orderable ?, Panoramas et synth`eses 14, Soci´et´e Math´ematique de France (2002). [6] P. Deligne, Les immeubles des groupes de tresses g´en´eralis´es, Inventiones Math. 17 (1972) 273-302. [7] D.M. Larue, On braid words and irreflexivity, Algebra Univ. 31 (1994) 104112. [8] J. Tits, Normalisateurs de tores. I: Groupes de Coxeter ´etendus, J. of Algebra 4 (1966) 96-116.