Systolic complexes and group presentations

TL;DR

This paper establishes conditions on group presentations that ensure their Cayley complexes are systolic, and applies these to classify Garside groups and analyze Artin groups, advancing understanding of their geometric properties.

Contribution

It introduces new conditions on group presentations that guarantee systolicity of Cayley complexes and applies these to classify Garside groups and analyze Artin groups.

Findings

Identified conditions ensuring Cayley complexes are systolic

Classified Garside groups with specific presentation properties

Determined when Artin groups' Cayley complexes are systolic

Abstract

We give conditions on a presentation of a group, which imply that its Cayley complex is simplicial and the flag complex of the Cayley complex is systolic. We then apply this to Garside groups and Artin groups. We give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions. We then also give a dual presentation for Artin groups and identify in which cases the flag complex of the Cayley complex is systolic.