Update: From Clinton's comment below follows that I made some mistakes (that I'm going to correct) and that the question is completely answered by Arzhantseva, Guba and Guyot. Besides giving a precise definition of what I meant with $\alpha(G)$, they proved that for any $n$, there is an $n$-generated amenable group with growth rate arbitrarily close to $2n-1$. About the very last question, it is also known that there are non-amenable semigroup with growth rate arbitrarily close to $1$. This means that there is probably no evident property which is shared.
Sometime in this topic I will not very precise - for instance, it will not clear if $\alpha(G)$ is well-defined (independent on the generating set); either it will not completely clear what is the exact meaning of growth rate $\leq r^n$. I hope the reader is not going to get angry: I' d like just to share some ideas for the moment, without being boring.
Warm-up question: for any real number $\geq1$, does there exist a finitely generated amenable group whose growth rate is $\geq r^n$?
For a finitely generated group $G$, let $\alpha(G)$ be its growth exponent, defined as the smallest real number $r>1$ such that the growth rate of $G$ is $\leq r^n$.
How is the notion of amenability distributed with respect to $\alpha$? I mean, it is clear that
- $\alpha(G)=1$, implies $G$ amenable
So the questions would be: does there exist $\alpha$ such that $\alpha(G)\leq\alpha$ if and only if $G$ is amenable? In case of negative answer, what happens for those $\alpha$'s for which there are both amenable and non-amenable groups? Are there any properties which are shared?
Does anyone have already studied the problem? References? Ideas?
Thanks in advance,