**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,


Valerio



On growth rate of finitely generated groups
I suppose I should convert the comment into an answer so that the question doesn't appear unanswered.


Given a group $G$ with finite generating set $S$, one can define its *rate of growth* (matching as much as possible the notation of the question) $\alpha(G,S)$ by
$$\alpha(G,S) = \lim\_{r \to \infty} \sqrt[r]{|B\_r|},$$
where $B\_r$ is the ball of radius $r$ about the identity in the Cayley graph $\mathrm{Cay}(G,S)$ of $G$ associated with $S$.


With this definition, if $\alpha(G,S) = 1$ (i.e., $G$ has subexponential growth), then $G$ is amenable. Also, if $\alpha(G,S) = 2|S| - 1$ and $|S|>1$, then $G$ is nonamenable (since in fact this only happens if $G$ is freely generated by $S$). However, there's no particular connection between rate of growth and amenability between these two extremes.


On the one hand, in [2] is exhibited for each $n>1$ a sequence of nonamenable groups on $n$ generators whose growth rates approach 1. On the other hand, In [1] is exhibited for each $n>1$ a sequence of amenable groups on $n$ generators whose growth rates approach $2n-1$.


[1] G.N. Arzhantseva, V.S. Guba, L. Guyot. Growth rates of amenable groups, Journal of Group Theory, 8 (2005), no.3, 389-394.


[2] R. Grigorchuk and P. de la Harpe. Limit behaviour of exponential growth rates for
finitely generated groups, Monographie de L’Enseignement Mathematique
38 (2001), 351-370.


