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

TL;DR

This paper establishes a Garside monoid structure for Artin-Tits groups of type  C, providing new insights into their algebraic properties and solving longstanding questions about their presentations and word problems.

Contribution

It introduces a Garside monoid presentation for Artin-Tits groups of type  C, extending known results from spherical types and addressing a general question in the field.

Findings

Proves the group is a group of fractions of a Garside monoid.

Provides a new presentation of the Artin-Tits group of type  C.

Implications for the word problem, centralizers, and center triviality.

Abstract

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.