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Next semester I will be teaching an introductory course on geometric group theory and there is a basic question that I do not know the answer to. Let $G$ be a finitely generated group with finite symmetric generating set $S$ and let $\Gamma$ be the corresponding Cayley graph. For each $n \geq 1$, let $B_{n}$ be the closed ball of radius $n$ in $\Gamma$ about the unit element $e$ and let $b_{n} = |B_{n}|$. Then it is known that $\lim b_{n}^{1/n}$ always exists. (For example, see de la Harpe's book.) My question is whether $\lim b_{n+1}/b_{n}$ always exists?

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up vote 29 down vote accepted

In the article

R. Grigorchuk and P. De La Harpe, On problems related to growth, entropy, and spectrum in group theory, Journal of Dynamical and Control Systems, Volume 3, Number 1, 51-89

on the lower part of page 58 the authors mention the manuscript

A. Machi, Growth functions and growth matrices for finitely generated groups. Unpublished manuscript, Univ. di Roma La Sapienza, 1986.

and explain an example due to Machi. Machi showed that the convergence of $b_{n+1}/b_n$ can fail for one generating set of ${\mathbb Z}_2 \star {\mathbb Z}_3$ and hold for another. In particular, the limit does not always exist. The two generating sets are $\lbrace s,t\rbrace$ and $\lbrace s,st\rbrace$, where $s$ and $t$ are the natural generators with $s^2=t^3=e$.

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Thanks very much! – Simon Thomas Aug 19 '10 at 22:02
Cool. A fun example to keep in mind (and to yourself!) the next time you teach a calculus class that the root test is stronger than the ratio test. – Pete L. Clark Aug 19 '10 at 23:29

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