# On growth rate of finitely generated groups

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?

Valerio

• I can not come up with an interpretation of $\alpha$ which makes the claim "$\alpha(G) > 2$ implies $G$ nonamenable" true, let alone clear. In any case, you might find the following paper useful: G.N. Arzhantseva, V.S. Guba, L.Guyot, Growth rates of amenable groups, Journal of Group Theory, 8 (2005), no.3, 389-394 Jun 30, 2011 at 18:42
• It seems to me that $\alpha(G)>2$ implies $|B(n+1)|\geq2|B(n)|$ for infinitely many $n$'s and this implies that $G$ is not amenable. But the latter step seems to contradict what they said in the paper.. so I'm certainly wrong, but this is quite against my intuition.. Jun 30, 2011 at 19:11
• I retagged with amenability since group-theory is so broad. Hope you don't mind! I'm not sure if there's a good tag for growth-rate kinds of questions. Perhaps "complexity-theory" but I didn't feel confident so I didn't add it. Jul 1, 2011 at 14:18
• No problem! thanks for making me discovering the existence of the amenability tag. Jul 3, 2011 at 0:19

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