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.

V2: Accepted in Bull. Lond. Math. Soc. Added discussion in the introduction on the various parts of the theorem, especially on the necessity of considering inverse monoids rather than semigroups (Remark 2.7), and on the notion of action (Remark 1.13). Several further clarifications and improvements to the exposition. 9 pages. Feedback is welcome!