On the complexity of the cogrowth sequence

Abstract

Given a finitely generated group with generating set S, we study the cogrowth sequence, which is the number of words of length n over the alphabet S that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from S. We prove that the cogrowth sequence is not P-recursive when G is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak. In addition, we compute the cogrowth for certain infinite families of free products of finite groups and free groups, and prove that a gap theorem holds: if S is a finite symmetric generating set for a group G and if a n denotes the number of words of length n over the alphabet S that are equal to 1 then either lim sup n a 1/n n ≤ 2 or lim sup n a 1/n n ≥ 2 √ 2.

Cogrowth Series for Free Products of Finite Groups

“…In §2.1 we work out several examples, which are listed in the statement of Theorem 1.2; The main result of §3 is a general bound on the degree of the minimial polynomial of co-growth series for the free products of finite groups, which gives Theorem 1.1 as a consequence. These results use ideas from free probability, which were suggested to two of the authors by one of the referees for the earlier paper [3]. In §4 we prove the gap result for radii of convergence given in Theorem 1.3, using the results from the preceding sections.…”

“…While there is a strong overlap between the various notions of complexity (group theoretic, language theoretic, and complexity of power series), there are nevertheless some families of groups which are typically regarded as being structurally well behaved whose corresponding cogrowth series are complex according to our notion of complexity. For example, it is shown in [3] that a finitely generated amenable group that is not virtually nilpotent can never have a generating series that satisfies a non-trivial homogeneous linear differential equation with rational function coefficients.…”