This is joint work with Miklos Abert, Nicolas Bergeron and Mikolaj Fraczyk. The growth of the sequence of Betti numbers is quite well understood when considering a suitable sequence of finite sheeted covers of a manifold or of finite index subgroups of a countable group.

We are interested in other homological invariants, like the growth of the mod $p$ Betti numbers and the growth of the torsion of the homology. We produce new vanishing results on the growth of torsion homologies in higher degrees for such groups as SL_d(Z), mapping class groups, Out(W_n), and Artin groups. As a by-product, we prove that the l²-Betti numbers of Out(W_n) vanish up to degree the integer part of n over 2 minus 1.

As a central tool, we introduce a quantitative homotopical method that constructs "small" classifying spaces for finite index subgroups, while controlling at the same time the complexity of the homotopy. Our method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups. I will present the basic objects and some of the ideas.