On growth rate of finitely generated groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:22:30Z http://mathoverflow.net/feeds/question/69207 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69207/on-growth-rate-of-finitely-generated-groups On growth rate of finitely generated groups Valerio Capraro 2011-06-30T17:41:15Z 2011-07-01T14:16:20Z <p><strong>Update:</strong> 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. </p> <p>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 <em>growth rate $\leq r^n$</em>. I hope the reader is not going to get angry: I' d like just to share some ideas for the moment, without being boring.</p> <p><strong>Warm-up question:</strong> for any real number $\geq1$, does there exist a finitely generated amenable group whose growth rate is $\geq r^n$?</p> <p>For a finitely generated group $G$, let $\alpha(G)$ be its <em>growth exponent</em>, defined as the smallest real number $r>1$ such that the growth rate of $G$ is $\leq r^n$.</p> <p>How is the notion of amenability distributed with respect to $\alpha$? I mean, it is clear that</p> <ul> <li>$\alpha(G)=1$, implies $G$ amenable</li> </ul> <p>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?</p> <p>Does anyone have already studied the problem? References? Ideas?</p> <p>Thanks in advance,</p> <p>Valerio</p> http://mathoverflow.net/questions/69207/on-growth-rate-of-finitely-generated-groups/69237#69237 Answer by Clinton Conley for On growth rate of finitely generated groups Clinton Conley 2011-07-01T07:27:45Z 2011-07-01T07:27:45Z <p>I suppose I should convert the comment into an answer so that the question doesn't appear unanswered.</p> <p>Given a group $G$ with finite generating set $S$, one can define its <em>rate of growth</em> (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 $S$.</p> <p>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.</p> <p>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 $2n-1$.</p> <p>[1] G.N. Arzhantseva, V.S. Guba, L. Guyot. Growth rates of amenable groups, Journal of Group Theory, 8 (2005), no.3, 389-394.</p> <p>[2] R. Grigorchuk and P. de la Harpe. Limit behaviour of exponential growth rates for finitely generated groups, Monographie de Lâ€™Enseignement Mathematique 38 (2001), 351-370.</p>