  Let N be an irreducible, compact 3-manifold with empty or toroidal boundary
which is not a closed graph manifold. Using recent work of Agol, Kahn-Markovic
and Przytycki-Wise we will show that pi_1(N) admits a cofinal filtration with
`fast' growth of Betti numbers as well as a cofinal filtration of pi_1(N) with
`slow' growth of ranks.
