Groups acting on semimetric spaces and quasi-isometries of monoids

TL;DR

This paper extends geometric group theory concepts to monoids and semigroups by studying group actions on asymmetric, partially-defined metric spaces and establishing quasi-isometry invariants for these algebraic structures.

Contribution

It introduces a notion of quasi-isometry for asymmetric spaces and extends the Svarc-Milnor Lemma to this setting, applying it to monoids and semigroups.

Findings

Monoids' properties are quasi-isometry invariants.

Extended Svarc-Milnor Lemma for asymmetric spaces.

Applicable to Cayley and Schutzenberger graphs.

Abstract

We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor Lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs; we apply our results to show a number of important properties of monoids are quasi-isometry invariants.