The focus of this chapter is on sequences of kleinian groups, typically sequences that are becoming degenerate in some way. For these, it is necessary to carefully distinguish between convergence of groups and convergence of quotient manifolds. The former has to do with sequences of groups whose generators converge, the latter with sequences of groups whose fundamental polyhedra converge. Our work in this chapter will enable us to describe the set of volumes of finite volume hyperbolic 3-manifolds. In preparation for this discussion, we will introduce the operation called Dehn surgery.

Algebraic convergence

In this section we will prove the two theorems which provide the basis for working with sequences of groups.

Let Γbe an abstract group and {φn : Γ → Gn} be a sequence of homomorphisms (also called representations) {φn} of Γ to groups Gn of Möbius transformations. Suppose for each γ ∈ Γ, limnn→∞ φn (γ) = φ(γ) exists as a Möbius transformation. Then the sequence {φn} is said to converge algebraically and its algebraic limit is the group G∞ = {φ(γ) : γ ∈ Γ}; φ : Γ → G∞ is a homomorphism. When we say a sequence of groups converges algebraically, we are assuming that behind the statement is a sequence of homomorphisms generating the sequence.

In particular, a sequence of r-generator groups Gn = 〈A1, nA2, n … Ar,n〉 is said to converge algebraically if Ak = limnn→ ∞Ak,n exists as a Möbius transformation, 1≤k ≤ r. Its algebraic limit is the group G = 〈A1, A2, … Ar〉.