Algebraic and geometric convergence

Abstract

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 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. We will continue on by introducing the operations called Dehn filling and surgery. This leads to the description of the set of volumes of finite volume hyperbolic 3-manifolds. Algebraic convergence In this section we will prove the two theorems which provide the basis for working with sequences of groups. Let Γ be an abstract group and { φ n : Γ → G n } be a sequence of homomorphisms (also called representations ) { φ n } of Γ to groups G n of Mobius transformations. Suppose for each γ ∈ Γ, lim n →∞ φ n (γ) = φ(γ) exists as a Mobius transformation. Then the sequence { φ n } is said to converge algebraically and its algebraic limit is the group of limits G ∞ = { φ(γ) : γ ∈ Γ}. When we say a sequence of groups converges algebraically, we are assuming that behind the statement is a sequence of homomorphisms or isomorphisms spawning the sequence. In particular, a sequence of r -generator groups G n = ⟨ A 1 , n , A 2 , n , … A r,n ⟩ is said to converge algebraically if A k = lim n →∞ A k ,n exists as a Mobius transformation, 1 ≤ k ≤ r . Its algebraic limit is the group G = ⟨ A 1 , A 2 , … A r ⟩. To make this terminology consistent with that used above, refer to the free group F r on r -generators and express G n as the sequence of representations φ n : F r → G n determined by sending the k -th generator of F r to A k,n . The convergence is controlled as spelled out by the following two fundamental results. Theorem 4.1.1 [Jorgensen 1976; Jorgensen and Klein 1982]. Let { G n } be a sequence of r-generator nonelementary kleinian groups converging algebraically to the group G. Then G is also a nonelementary kleinian group, and the map A k → A k,n , 1 ≤ k ≤ r, determines a homomorphism ϕ n : G → Gn for all large indices n . In general ϕ n will not be an isomorphism.