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

Let G be a group generated by $r$ elements $g_1,g_2,\mathrm{.}\mathrm{.}\mathrm{.},g_r\mathrm{.}$ Among the reduced words in $g_1,g_2,\mathrm{.}\mathrm{.}\mathrm{.},g_r$ of length $n$ some, say ${\gamma }_n,$ represent the identity element of the group $G\mathrm{.}$ 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 $g_1,g_2,\mathrm{.}\mathrm{.}\mathrm{.},g_r\mathrm{.}$ We show by analytic methods that the numbers ${\gamma }_n$ vary regularly; i.e. the ratio ${\gamma }_{2n+2}\mathrm{/}{\gamma }_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\gamma (z),$ the generating function associated with the coefficients ${\gamma }_n\mathrm{.}$