Narrow Results

Let F denote the Thompson group with standard generators A = x₀, B = x₁. It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to

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 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 $G$ 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 $S$ 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 ${lim sup}_n{a}_n^{1/n}$ is either $1$, $2$ or at least $2\sqrt{2}$.