Groups acting on semimetric spaces and quasi-isometries of monoids

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 Švarc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schützenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.