Classifying spaces from Ore categories with Garside families

TL;DR

This paper introduces a method to construct classifying spaces for groups derived from Ore categories with Garside families, generalizing known complexes for braid and Thompson groups, and demonstrating the finiteness properties of these groups.

Contribution

It provides a unified construction of classifying spaces for groups from Ore categories with Garside structures, extending existing complexes and introducing new categories via Zappa–Szép products.

Findings

Constructed classifying spaces generalizing Brady's and Stein–Farley's complexes

Proved Garside groups have finite classifying spaces

Introduced the group Braided T and showed it is of type F_infinity

Abstract

We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Sz\'ep product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group Braided T and show that it is of type $F_\infty$.