Abstract

We study the connection between
amenability, Følner conditions and the geometry of
finitely generated semigroups. Using results of Klawe, we
show that within an extremely broad class of semigroups
(encompassing all groups, left cancellative semigroups,
finite semigroups, compact topological semigroups, inverse
semigroups, regular semigroups, commutative semigroups and
semigroups with a left, right or two-sided zero element),
left amenability coincides with the strong Følner
condition. Within the same class, we show that a finitely
generated semigroup of subexponential growth is left
amenable if and only if it is left reversible. We show
that the (weak) Følner condition is a left
quasi-isometry invariant of finitely 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 Følner condition, in
terms of the existence of weak Følner sets
satisfying a local injectivity condition on the relevant
translation action of the semigroup.