Garside categories, periodic loops and cyclic sets

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 Bökstedt-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

33 pages. First abridged version

References in corpus (3)

• Non-positively curved aspects of Artin groups of finite type
• A class of Garside groupoid structures on the pure braid group
• Conjugacy in Garside Groups III: Periodic braids

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• Finite complex reflection arrangements are K(pi,1)
• Helly meets Garside and Artin
• Periodic elements in Garside groups
• Algorithms for Garside calculus
• Left-Garside categories, self-distributivity, and braids
• Totally normal cellular stratified spaces and applications to the configuration space of graphs
• Groupes de réflexion, géométrie du discriminant et partitions non-croisées
• Basic results on braid groups
• Notes on periodic elements of Garside groups
• Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology
• Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups