A short proof of the Grigorchuk-Cohen cogrowth theorem

Abstract

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