In this dissertation we study a family of two-relator groups which contains surface-plus-one-relation groups, that is one-relator quotients of fundamental groups of surfaces. In particular we finish computing the $L^2$-Betti numbers of surface-plus-one-relation groups. The non-orientable case is new, and the orientable case was obtained by Warren Dicks and Peter Linnell.
We study this family of two-relation groups algebraically, and we develop a Bass-Serre theoretical version of theorems about Cohen-Lyndon asphericity and quotients of locally indicable groups. In particular we have a new pure algebraic version of Howie's Induction method.
Among other results, for a group G in this family of two-relation groups we are able to describe the torsion subgroups of G, we obtain a classifying space of proper actions of G, we find exact sequences of G-modules, and we compute the Bredon homology of G. Such a group is of type VFL and has virtual cohomological dimension less or equal to 2.
The calculation of the $L^2$-Betti numbers, uses a 1979 result of Dunwoody, and, in another direction, the thesis describes further applications of Dunwoody's 1979 result to simplify and unify proofs of many results about virtually free groups that previously invoked Dunwoody's 1985 Accessibility Theorem.