Rank gradient, cost of groups and the rank versus Heegaard genus problem

Abstract. We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups Gn < G of increasing finite index in a fixed finitely generated group G, assuming n Gn = 1. We focus in particular on finitely presented residually free groups, calculating their ℓ 2 betti numbers, rank gradient and asymptotic deficiency.If G is a limit group and K is any field, then for all j ≥ 1 the limit of dim H j (Gn, K)/[G, Gn] as n → ∞ exists and is zero except for j = 1, where it equals −χ(G). We prove a homotopical version of this theorem in which the dimension of dim H j (Gn, K) is replaced by the minimal number of j-cells in a K(Gn, 1); this includes a calculation of the rank gradient and the asymptotic deficiency of G. Both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of slow groups.We prove that if a residually free group G is of type FPm but not of type FP∞, then there exists an exhausting filtration by normal subgroups of finite index Gn so that limn dim H j (Gn, K)/[G : Gn] = 0 for j ≤ m. If G is of type FP∞, then the limit exists in all dimensions and we calculate it.

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