The ratio and generating function of cogrowth coefficients of finitely generated groups
Abstract
Description
Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $γ_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $γ_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $γ_n$ vary regularly; i.e. the ratio $γ_{2n+2}/γ_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $γ(z),$ the generating function associated with the coefficients $γ_n.$