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?



On the size of balls in Cayley graphs
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](http://www.springerlink.com/content/475575746034r344/)


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 Unpublished manuscript, Univ.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 $s^2=t^3=e$.


