Density of growth-rates of subgroups of a free group.

We prove that the set of growth-rates of subgroups of a rank 2 free group is dense in [1,3]. Our main tool is the probabilistic method: we prove that for any \alpha \in [1,3], there exists a sequence of distributions {P_N} where P_N is a distribution over graphs with N vertices such that
1. For any N and any graph G in the support of P_N, every node in G has either degree 2 or 4, and
2. Pr[gr(G) = \alpha + o(N)] >= 1-o(N), where gr(G) is the growth rate of the universal cover of G.
This is a joint work with Mixalis Louvaris and Daniel Wise.