A note on the growth of Betti numbers and ranks of 3-manifold groups

Let N   be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. We show that it follows from the work of Agol, Kahn–Markovic and Przytycki–Wise that _π1(N)${\pi }_1(N)$ admits a cofinal filtration with ‘fast’ growth of Betti numbers as well as a cofinal filtration of _π1(N)${\pi }_1(N)$ with ‘slow’ growth of ranks.