Amenability and Geometry of Semigroups

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.