A short proof of the Grigorchuk-Cohen cogrowth theorem

Abstract:

Let $G$ be a group generated by ${g_1}, \ldots ,{g_r}$. There are exactly $2r{(2r - 1)^{n - 1}}$ reduced words in ${g_1}, \ldots ,{g_r}$ of length $n$. Part of them, say ${\gamma _n}$ represents identity element of $G$. Let $\gamma = \lim \sup \gamma _n^{1/n}$. We give a short proof of the theorem of Grigorchuk and Cohen which states that $G$ is amenable if and only if $\gamma = 2r - 12$. Moreover we derive some new properties of the generating function $\sum {{\gamma _n}{z^n}}$.