Does there exists a finitely presented group of exponential growth which does not contain free sub-semigroups (of rank $\geq 2$)?
finitely presented groups of exponential growth
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group-theory
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1Steve 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. – 2012-04-23