The Subgroup Growth Spectrum of Virtually Free Groups

For a finitely generated group Γ denote by μ(Γ) the growth coefficient of Γ, that is, the infimum over all real numbers d such that sn(Γ) < n!d. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. For a finitely generated group Γ denote by sn(Γ) the number of subgroups of index n in Γ, and define the growth coefficient μ(Γ) as μ(Γ) = lim sup n→∞ log sn(Γ) n log n . This quantity has been computed in a variety of cases, including free product of finite groups [5], Fuchsian groups [6], and certain one-relator groups [7]. In all these examples, μ(Γ) can be expressed as −1 + ∑k i=1(1 − 1 ni ) for certain integers ni ≥ 2. This observation led to the question as to which values μ(Γ) can attain; in particular, in [8] it was asked whether there exists a sequence of groups (Γn), such that μ(Γn) converges from above. This question is answered by the following Theorem. Theorem 1. The set {μ(Γ) : Γ finitely generated, virtually free} is equal to the set of non-negative rational numbers. The proof of this result falls into two steps, the first one being the construction of virtually free groups with given growth coefficient, these groups are described in the following theorem. Theorem 2. Let k, ` be integers, p a prime number, and suppose that k − `p ≥ 2. Define the group Γp,k,` as the amalgamated product Sk ∗Cp Sk, where a generator of Cp is mapped onto an element consisting of ` cycles of length p in both groups. Similarly, for p odd or p = 2 and ` even, let Γ+p,k,` be the amalgamated product Ak ∗Cp Ak. Then we have μ(Γp,k,`) = 1− (p− 1)`+ 1 + δ k , where δ = 1 if p = 2 and ` is odd, and δ = 0 otherwise, apart from the exceptions (p, k, `) = (2, 5, 2), (3, 7, 2), for which we have μ(Γ2,5,2) = 12 and μ(Γ3,7,2) = 2 5 , and μ(Γ+p,k,`) = 1− (p− 1)`+ 2 k with the exceptions μ(Γ2,5,2) = 1 3 and μ(Γ + 3,7,2) = 1 3 . In a second step we have to show that the growth coefficient of any virtually free group is rational. This is implied by the following result.