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

الأصل · EN

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.