A Garside presentation for Artin-Tits groups of type ${\tilde{C}}_n$

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.