## Abstract

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.

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 language | English |
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Pages (from-to) | 8087-8103 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Early online date | 1 May 2017 |

DOIs | |

Publication status | Published - 1 May 2017 |