We use cohomological group theory and properties of -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.
Jonathan A. HILLMAN. "Tits alternatives and low dimensional topology." J. Math. Soc. Japan 55 (2) 365 - 383, April, 2003. https://doi.org/10.2969/jmsj/1191419121