Groups acting on semimetric spaces and quasi-isometries of monoids

Robert Gray*, Mark Kambites

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    8 Citations (Scopus)


    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 that a number of important properties of monoids are quasi-isometry invariants.

    Original languageEnglish
    Article numberPII S0002-9947(2012)05868-5
    Pages (from-to)555-578
    Number of pages24
    JournalTransactions of the American Mathematical Society
    Issue number2
    Publication statusPublished - Feb 2013


    • Monoid
    • group
    • finitely generated
    • action
    • semimetric space
    • quasi-metric space
    • Des Demi-Groupes
    • Finiteness conditions
    • Semigroups
    • Growth
    • Ends
    • Inverse


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