Volume gradients and homology in towers of residually-free groups

Martin R. Bridson and Dessislava H. Kochloukova

We study the asymptotic growth of homology groups and the cellular volume of
classifying spaces as one passes to normal subgroups G_n of increasing finite
index in a fixed finitely generated group G, assuming that the intersection of
the G_n is trivial. We focus in particular on finitely presented residually free
groups, calculatint their \ell_2 betti numbers, rank gradient and asymptotic

If $G$ is a limit group and $K$ is any field, then for all $j\ge 1$ the
limit of $\dim H_j(G_n,K)/[G,G_n]$ as $n\to\infty$ exists and is zero except
for $j=1$, where it equals $-\chi(G)$.

We prove a homotopical version of this last result in which the dimension of
$\dim H_j(G_n,K)$ is replaced by the minimal number of $j$-cells in a $K(G_n,1)$; 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 {\em{slow}} groups.

We prove that if a residually free group $G$ is of type $\rm{FP}_m$ but not of
type $\rm{FP}_{\infty}$, then there exists an exhausting filtration by normal subgroups of finite
index $G_n$ so that $\lim_n \dim H_j (G_n, K) / [G : G_n] = 0 \hbox{ for } j \leq m$.
If $G$ is of type $\rm{FP}_{\infty}$, then the limit exists in all dimensions and we calculate it.

Submitted August 2013.

32 pages, no figures.