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Does there exists a finitely presented group of exponential growth which does not contain free sub-semigroups (of rank $\geq 2$)?

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I think this is a difficult question. You could try asking it on MathOverflow. A good candidate would be a finitely presented infinite torsion group, because that would certainly not contain a free subsemigroup, and it would have a good chance of having exponential growth. The problem with that is that it is still an open question whether such groups exist, although it is widely believed that they do. –  Derek Holt Apr 22 '12 at 11:02
    
I don't know anything about free semigroups, but what about a solvable group that is not virtually nilpotent? Something like $\langle a,b,c\ |\ [a,b]=1, a^c=a^2b, b^c=ab\rangle$. Just seems like "solvable" and "free" shouldn't mix. –  user641 Apr 22 '12 at 16:52
    
I guess this group contains the lamplighter group and hence a free semi-group. –  Mustafa Gokhan Benli Apr 23 '12 at 9:25
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Let me also add that for elementary amenable groups exponential growth is equivalent to containing free semigroups. –  Mustafa Gokhan Benli Apr 23 '12 at 10:34
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Steve D: I convinced myself that $c^2$ and $c^2a$ generate a free semigroup in your example. I suspect that examples like this will all turn out to have free subsemigroups. –  Derek Holt Apr 23 '12 at 11:00
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