Amenable groups not containing free semigroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:13:43Z http://mathoverflow.net/feeds/question/56778 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56778/amenable-groups-not-containing-free-semigroups Amenable groups not containing free semigroups Mustafa Gokhan Benli 2011-02-27T00:29:14Z 2011-02-27T05:33:22Z <p>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?</p> http://mathoverflow.net/questions/56778/amenable-groups-not-containing-free-semigroups/56793#56793 Answer by Denis Osin for Amenable groups not containing free semigroups Denis Osin 2011-02-27T05:33:22Z 2011-02-27T05:33:22Z <p>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. </p> <p>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].</p>