Garside categories, periodic loops and cyclic sets

TL;DR

This paper explores the structure of Garside groupoids, showing how periodic loops relate to Garside elements, and introduces divided Garside categories to analyze their properties, including applications to complex reflection arrangements.

Contribution

It introduces divided Garside categories as a new tool and demonstrates their use in understanding periodic elements and the $K( extpi,1)$ property for reflection arrangements.

Findings

Periodic loops can be viewed as Garside elements in related Garside groupoids.

Centralisers of periodic elements are weak Garside groups.

Divided Garside categories generalize cyclic category subdivisions.

Abstract

Garside groupoids, as recently introduced by Krammer, generalise Garside groups. A weak Garside group is a group that is equivalent as a category to a Garside groupoid. We show that any periodic loop in a Garside groupoid $\CG$ may be viewed as a Garside element for a certain Garside structure on another Garside groupoid $\CG_m$, which is equivalent as a category to $\CG$. As a consequence, the centraliser of a periodic element in a weak Garside group is a weak Garside group. Our main tool is the notion of divided Garside categories, an analog for Garside categories of B\"okstedt-Hsiang-Madsen's subdivisions of Connes' cyclic category. This tool is used in our separate proof of the $K(\pi,1)$ property for complex reflection arrangements