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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asked Feb 27, 2011 at 0:29
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1$\begingroup$ 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?'. $\endgroup$– HJRWCommented Feb 27, 2011 at 1:41
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1 Answer
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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 non-cyclic free subsemigroups is equivalent to exponential growth [C. Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), 3, 396-407].
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$\begingroup$ Thank you I was not aware of this result. $\endgroup$ Commented Feb 28, 2011 at 6:04