Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups

TL;DR

This paper generalizes the Dehornoy-Lafont order complex to categories, enabling efficient homology computations for Garside groups and applying this to study the homology of exceptional complex braid groups, including the Borchardt braid group.

Contribution

It introduces a categorical generalization of the order complex and provides computational techniques to analyze the homology of complex braid groups.

Findings

Completed homology results for exceptional complex braid groups

Developed methods to reduce computational time in homology calculations

Analyzed the Borchardt braid group using the new categorical framework

Abstract

The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group $B(G_{31})$ through its associated Garside category.