  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 $\gamma_n.$
