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 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 is a Coxeter group, one can consider the length on with respect to the generating set consisting of all reflections. Let be a Coxeter element in and let be the set of elements such that can be written with . We define the monoid to be the monoid generated by a set in one-to-one correspondence, , with with only relations whenever , and are in and . We conjecture that the group of quotients of is the Artin--Tits group associated to 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 . 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 .
Cite this article
François Digne, Présentations duales des groupes de tresses de type affine <em>Ã</em>. Comment. Math. Helv. 81 (2006), no. 1, pp. 23–47DOI 10.4171/CMH/41