Parabolic subgroups of Garside groups II: ribbons

TL;DR

This paper introduces the ribbon groupoid for Garside groups, proves it forms a Garside groupoid under certain conditions, and decomposes it into a semi-direct product of parabolic subgroupoids, with applications to Artin-Tits groups.

Contribution

It defines the ribbon groupoid for Garside groups, proves its Garside structure under a technical hypothesis, and provides a decomposition into parabolic subgroupoids with a detailed presentation.

Findings

The ribbon groupoid is a Garside groupoid under certain conditions.

Decomposition of the groupoid into a semi-direct product of parabolic subgroupoids.

Description of quasi-centralizers in Garside groups.

Abstract

We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type.