Simplicial pseudohyperplane arrangements give weak Garside groups

TL;DR

This paper establishes that the fundamental groups of complexified pseudohyperplane arrangements and Salvetti complexes of oriented matroids are examples of weak Garside groups, connecting combinatorial and topological structures.

Contribution

It introduces new classes of weak Garside groups derived from pseudohyperplane arrangements and oriented matroids, linking algebraic, combinatorial, and topological frameworks.

Findings

Fundamental groups of complexified pseudohyperplane arrangements are weak Garside groups.

Salvetti complexes of oriented matroids have fundamental groups that are weak Garside groups.

Provides new examples connecting Garside theory with combinatorial topology.

Abstract

In this note we connect the language of Bessis's Garisde categories with Salvetti's metrical-hemisphere complexes in order to find new examples of weak Garside groups. As our main example, we show that the fundamental group of the (appropriately defined) complexified complement of a pseudohyperplane arrangement is a weak Garside group. As a consequence of the Folkman-Lawrence topological realization theorem, we also show that fundamental group of the Salvetti complex of a ("simplicial") oriented matroid is a weak Garside group. This provides novel examples of weak Garside groups.