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Dual presentations of braid groups of affine type $$\widetilde A$$. (Présentations duales des groupes de tresses de type affine $$\widetilde A$$.) (French) Zbl 1143.20020
Summary: 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 $$\widetilde A_n$$ 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 $$\widetilde A$$. 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 $$A$$.

MSC:
 20F36 Braid groups; Artin groups 20F05 Generators, relations, and presentations of groups 20M05 Free semigroups, generators and relations, word problems 20F55 Reflection and Coxeter groups (group-theoretic aspects) 57M07 Topological methods in group theory
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