Tits alternatives and low dimensional topology

Abstract

We use cohomological group theory and properties of $L^2$-Betti numbers to determine the solvable groups with presentations of deficiency 1, to give a new proof of the "Tits alternative" for subgroups of Haken 3-manifold groups, and to study the fundamental groups of closed 4-manifolds with Euler characteristic 0 and, in particular, 2-knot groups.