Cogrowth series for free products of finite groups

Abstract

Given a finitely generated group with generating set [Formula: see text], we study the cogrowth sequence, which is the number of words of length [Formula: see text] over the alphabet [Formula: see text] that are equal to the identity in the group. This is related to the probability of return for walks on the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when [Formula: see text] has a finite-index-free subgroup (using a result of Dunwoody). In this work, we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if [Formula: see text] is a finite symmetric generating set for a group [Formula: see text] and if [Formula: see text] denotes the number of words of length [Formula: see text] over the alphabet [Formula: see text] that are equal to [Formula: see text] then [Formula: see text] is either [Formula: see text], [Formula: see text] or at least [Formula: see text].