Geometric Inverse Semigroup Theory: a note on the Milnor-Schwarz Lemma for inverse monoids

TL;DR

This paper extends the Milnor-Schwarz lemma to inverse monoids acting on presheaves of geodesic metric spaces, providing two proofs—one elementary and one using Vietoris-Rips complexes.

Contribution

It generalizes a fundamental geometric group theory result to inverse monoids and introduces two distinct proofs, including one based on Vietoris-Rips complexes.

Findings

The Milnor-Schwarz lemma is valid for inverse monoids acting on presheaves of geodesic metric spaces.

Two proofs are provided: an elementary one and another involving Vietoris-Rips complexes.

The work connects inverse monoid actions with geometric and topological methods.

Abstract

We generalise the Milnor-Schwarz lemma to inverse monoids acting on presheaves of geodesic metric spaces. We provide two proofs of this fact: one only uses elementary techniques, inspired by the arguments for group actions on metric spaces; the other involves a version of the Vietoris-Rips complex, and builds on work of Chung-Mart\'inez-Szak\'acs.