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

Let G be a group generated by r elements g1, g2, . . . , gr. Among the reduced words in g1, g2, . . . , gr of length n some, say γn, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ2n has a limit, called the cogrowth exponent with respect to generators g1, g2, . . . , gr. We show by analytic methods that the numbers γn vary regularly; i.e. the ratio γ2n+2/γ2n is also convergent. Moreover we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients γn. Every groupG generated by r elements can be realized as a quotient of the free group IFr on r generators by a normal subgroup N of IFr, in such a way that the generators of the free group IFr are sent to the generators of the group G. With the set of generators of IFr we associate the length function of words in these generators. The cogrowth coefficients γn = #{x ∈ N | |x| = n} were first introduced by Grigorchuk in [2]. The numbers γn measure how big the group G is when compared with IFr. It has been shown that the quantities 2n √ γ2n have a limit denoted by γ, and called the growth exponent of N in IFr. Since the subgroup N can have at most 2r(2r−1)n−1 elements of length n, the cogrowth exponent γ can be at most 2r − 1. The famous Grigorchuk result, proved independently by J. M. Cohen in [1], states that the group G is amenable if and only if γ = 2r − 1 (see also [6], [8]). ∗This work has been partially supported by KBN (Poland) under grant 2 P03A 030 09. 1991 Mathematics Subject Classification. Primary 20F05, 20E05