Pr\'esentations duales des groupes de tresses de type affine $\tilde A$

F. Digne

Abstract.
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. These conjectures
are known to be true for spherical type Artin-Tits groups. In the present
paper we prove them for Artin-Tits groups of type $\tilde A$. Moreover, for a suitable choice of the Coxeter element we obtain a (quasi-)
Garside monoid, which allows to define normal forms on the Artin-Tits group and to solve some questions such
as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group.