Dual presentations of braid groups of affine type \(\widetilde A\).
(Présentations duales des groupes de tresses de type affine \(\widetilde A\).)

*(French)*Zbl 1143.20020Summary: 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 \(A\).

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 |