**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

deficiency.

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.