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?
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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. For elementary amenable (in particular, solvable) groups, existence of noncyclic free subsemigroups is equivalent to exponential growth [C. Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), 3, 396407]. 

