The key algebraic idea used in this book is to study the homology of covering spaces as modules over the group ring of the group of covering transformations. In this chapter we shall summarize the relevant notions from group theory: elementary amenable groups, finiteness conditions, the stable invariant basis number property, and the connection between ends and the vanishing of cohomology with coefficients in a free module.

Our principal references for group theory are [Bi], [DD] and [Ro].

Group theoretic notation and terminology

Let G be a group. Then G′ and ζG denote the commutator subgroup and centre of G, respectively. The outer automorphism group of G is Out(G) = Aut(G)/Inn(G), where Inn(G) ≅ G/ζG is the subgroup of Aut(G) consiting of conjugations by elements of G. If H is a subgroup of G let NG(H) and CG(H) denote the normalizer and centralizer of H in G, respectively.

If p : G → Q is an epimorphism with kernel N we shall say that G is an extension of Q = G/N by the normal subgroup N. The action of G on N by conjugation determines a homomorphism from G/N to Out(N) = Aut(N)/Inn(N). If G/N ≅ Z the extension splits: a choice of element t in G which projects to a generator of G/N determines a right inverse to p.