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

Let G be a group generated by r elements ${\displaystyle g_1,\dots ,g_r}$. Among the reduced words in ${\displaystyle g_1,\dots ,g_r}$ of length n some, say ${\displaystyle {\gamma }_n}$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\displaystyle {\gamma }_{2n}}$ has a limit, called the cogrowth exponent with respect to the generators ${\displaystyle g_1,\dots ,g_r}$. We show by analytic methods that the numbers ${\displaystyle {\gamma }_n}$ vary regularly, i.e. the ratio ${\displaystyle \frac{{\gamma }_{2n+2}}{{\gamma }_{2n}}}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients ${\displaystyle {\gamma }_n}$.