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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 $r\in(1,2)$, \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$amenable •$\alpha(G)>2$, implies$G$non amenable So the questions would be: does there exist$\alpha$such that$\alpha(G)\leq\alpha$if and only if$G$is amenable? In case of negative answer, what happens for those$\alpha$'s for which there are both amenable and non-amenable groups? Are there any properties which are shared? Does anyone have already studied the problem? References? Ideas? Thanks in advance, Valerio 2 added 19 characters in body Sometime in this topic I will not very precise - for instance, it will not clear if$\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$mean. I hope the reader is not going to get angry: I' d like just to share some ideas for the moment, without being boring. Warm-up question: for any real number$r\in(1,2)$, does there exist a finitely generated amenable group whose growth rate is$\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$amenable •$\alpha(G)>2$, implies$G$non amenable So the questions would be: does there exist$\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?