A Garside presentation for Artin-Tits groups of type $\tilde C_n$

François Digne (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the ''generated group'' method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde C$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type $\tilde A$ under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

Mots clés: Braids; Garside; Artin-Tits groups; affine Coxeter groups

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

François Digne. A Garside presentation for Artin-Tits groups of type $\tilde C_n$. Annales de l'Institut Fourier, 2012, 62 (2), pp.641-666. ⟨hal-00459124v2⟩