A Svarc-Milnor lemma for monoids acting by isometric embeddings

TL;DR

This paper extends the Svarc-Milnor Lemma to monoids acting by isometric embeddings on spaces with asymmetric, partially-defined distances, broadening geometric group theory techniques to semigroup theory.

Contribution

It introduces a novel extension of the Svarc-Milnor Lemma applicable to monoids acting on asymmetric metric spaces, advancing the understanding of monoid actions in geometric contexts.

Findings

Extended the Svarc-Milnor Lemma to monoids with isometric actions

Established a framework for monoid actions on asymmetric metric spaces

Provided examples of monoid actions analogous to group actions in geometric group theory

Abstract

We continue our programme of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially-defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Svarc-Milnor Lemma to this setting.