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

Giorgio Mangioni, Francesco Tesolin

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-Martínez-Szakács.

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

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-Martínez-Szakács.

Paper Structure

This paper contains 3 sections, 13 theorems, 26 equations.

Table of Contents

  1. Background on inverse semigroups
  2. Milnor-Schwarz for inverse monoids
  3. Comparison to work of Chung-Martínez-Szakács

Key Result

Theorem 1

Let $S$ be an inverse monoid, with identity element $1$, and $(X, E(S),p)$ be a presheaf of geodesic metric spaces. If $S$ acts on $X$ properly and coboundedly then:

Theorems & Definitions (40)

  • Theorem 1
  • Definition 1.1: Inverse semigroup, domain and range
  • Definition 1.2: Idempotents
  • Remark 1.3
  • Definition 1.4: Partial order
  • Example 1.5: Partial bijections
  • Remark 1.6
  • Definition 1.7: Congruence and Green's relations
  • Lemma 1.8
  • proof
  • ...and 30 more