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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asked Aug 19, 2010 at 21:08
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In the article
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 Unpublished manuscript, Univ.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 $s^2=t^3=e$.
answered Aug 19, 2010 at 21:53
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5$\begingroup$ 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. $\endgroup$ Aug 19, 2010 at 23:29
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$\begingroup$ The link to Grigorchuk and de la Harpe's paper at
springerlink.comis broken, but the article can be found at doi:10.1007/BF02471762 (Zbl 0949.20033). $\endgroup$ Sep 11, 2022 at 12:14