On the size of balls in Cayley graphs - MathOverflow 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 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 Answer by Andreas Thom for On the size of balls in Cayley graphs 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 Unpublished manuscript, Univ.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>