Approximation by Maximal Cusps in Boundaries of Deformation Spaces of Kleinian Groups

Abstract

Let M be a compact, oriented, irreducible, atoroidal 3-manifold with nonempty boundary. Let CC₀(M) denote the space of convex cocompact Kleinian groups uniformizing M. We show that any Kleinian group in the boundary of CC₀(M) whose limit set is the whole sphere can be approximated by maximal cusps. Density of maximal cusps on the boundary of Schottky space is derived as a corollary. We further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3-manifold with connected conformal boundary.