Présentations duales des groupes de tresses de type affine <em>Ã</em>

Abstract

Artin--Tits groups of spherical type have two well-known Garside structures, coming respectively from the divisibility properties of the classical Artin monoid and of the dual monoid. For general Artin--Tits groups, the classical monoids have no such Garside property. In the present paper we define dual monoids for all Artin--Tits groups and we prove that for the type $\tilde A_n$ we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin--Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin--Tits group. More precisely, if $W$ is a Coxeter group, one can consider the length $l_R$ on $W$ with respect to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter element in $W$ and let $P_c$ be the set of elements $p\in W$ such that $c$ can be written $c=pp'$ with $l_R(c)=l_R(p)+l_R(p')$. We define the monoid $M(P_c)$ to be the monoid generated by a set $\underline P_c$ in one-to-one correspondence, $p\mapsto \underline p$, with $P_c$ with only relations $\underline{pp'}=\underline p.\underline p'$ whenever $p$, $p'$ and $pp'$ are in $P_c$ and $l_R(pp')=l_R(p)+l_R(p')$. We conjecture that the group of quotients of $M(P_c)$ is the Artin--Tits group associated to $W$ and that it has a simple presentation (see 1.1 (ii)). These conjectures are known to be true for spherical type Artin--Tits groups. Here we prove them for Artin--Tits groups of type $\tilde A$. Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type $A$.