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

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

2007-11-22

arxiv.org/abs/0711.3514v1

Studia Mathematica 131 (1998), 89-94

Mathematics

Functional Analysis

Scientific paper

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 $\gamma_n.$

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