most recent 30 from http://mathoverflow.net 2013-06-19T17:08:02Z http://mathoverflow.net/feeds/question/36126 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36126/on-the-size-of-balls-in-cayley-graphs Simon Thomas 2010-08-19T21:08:44Z 2010-08-19T21:53:25Z

<p>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?</p>

http://mathoverflow.net/questions/36126/on-the-size-of-balls-in-cayley-graphs/36132#36132 Andreas Thom 2010-08-19T21:53:25Z 2010-08-19T21:53:25Z

<p>In the article </p> <p><a href="http://www.springerlink.com/content/475575746034r344/" rel="nofollow">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 </a></p> <p>on the lower part of page 58 the authors mention the manuscript</p> <p>A. Machi, Growth functions and growth matrices for finitely generated groups. Unpublished manuscript, Univ. di Roma La Sapienza, 1986.</p> <p>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 $s^2=t^3=e$.</p>