The ratio and generating function of cogrowth coefficients of finitely generated groups

TL;DR

This paper investigates the asymptotic behavior of cogrowth coefficients in finitely generated groups, establishing their regular variation and analyzing the analytic properties of their generating functions.

Contribution

It introduces analytic methods to study the regularity and limits of cogrowth coefficients and characterizes the domain of holomorphy of their generating functions.

Findings

The ratio _{2n+2}/_{2n} converges.

The 2n-th root of _{2n} has a well-defined limit.

The generating function (z) has a precisely characterized domain of holomorphy.

Abstract

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 $\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $\gamma_{2n}$ 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 $\gamma_n$ vary regularly; i.e. the ratio $\gamma_{2n+2}/\gamma_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\gamma(z),$ the generating function associated with the coefficients $\gamma_n.$