Amenability and Geometry of Semigroups

Robert Gray, Mark Kambites

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    We study the connection between amenability, Flner con-
    ditions and the geometry of nitely generated semigroups. Using re-
    sults of Klawe, we show that within an extremely broad class of semi-
    groups (encompassing all groups, left cancellative semigroups, nite
    semigroups, compact topological semigroups, inverse semigroups, reg-
    ular semigroups, commutative semigroups and semigroups with a left,
    right or two-sided zero element), left amenability coincides with the
    strong Flner condition. Within the same class, we show that a nitely
    generated semigroup of subexponential growth is left amenable if and
    only if it is left reversible. We show that the (weak) Flner condition is a
    left quasi-isometry invariant of nitely generated semigroups, and hence
    that left amenability is a left quasi-isometry invariant of left cancellative
    semigroups. We also give a new characterisation of the strong Flner
    condition, in terms of the existence of weak Flner sets satisfying a local
    injectivity condition on the relevant translation action of the semigroup.
    Original languageEnglish
    Pages (from-to)8087-8103
    JournalTransactions of the American Mathematical Society
    Early online date1 May 2017
    Publication statusPublished - 1 May 2017


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