# Amenable groups not containing free semigroups

It is known that all amenable groups do not contain free subgroups (of rank>1). But there are amenable groups containing free semigroups. Which amenable groups cannot contain free semigroups?

-
Well, obviously a group containing a free semigroup has exponential growth. So a more specific question might be 'Is there an example of an amenable group with exponential growth but no free sub-semigroup?'. –  HJRW Feb 27 '11 at 1:41

This is the answer to the question asked by Henry. The wreath product $\mathbb Z_2 {\rm wr} G$, where $G$ is the Grigorchuk (torsion) group of subexponential growth, obviously has exponential growth and is amenable and torsion. In particular, it has no free subsemigroups.