Volume gradients and homology in towers of residually-free groups - Bridson, MR; Kochloukova, DH (Springer, 2016)
We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups (Formula presented.) of increasing finite index in a fixed finitely generated group G, assuming (Formula presented.). We focus in particular on finitely presented residually free groups, calculating their (Formula presented.) betti numbers, rank gradient and asymptotic deficiency. If G is a limit group and K is any field, then for all (Formula presented.) the limit of (Formula presented.) as (Formula presented.) exists and is zero except for (Formula presented.), where it equals (Formula presented.). We prove a homotopical version of this theorem in which the dimension of (Formula presented.) is replaced by the minimal number of j-cells in a (Formula presented.); 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 (Formula presented.) but not of type (Formula presented.), then there exists an exhausting filtration by normal subgroups of finite index (Formula presented.) so that (Formula presented.). If G is of type (Formula presented.), then the limit exists in all dimensions and we calculate it.