Hyperbolic limits of cantor set complements in the sphere

Abstract

Let M $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in S 3 $\mathbb {S}^3$ . Then, if M $M$ admits an exhaustion by π 1 $\pi _1$ -injective sub-manifolds there exists Cantor sets C n ⊆ S 3 $C_n\subseteq \mathbb {S}^3$ such that N n = S 3 ∖ C n $N_n=\mathbb {S}^3\setminus C_n$ is hyperbolic and N n → M $N_n\rightarrow M$ geometrically.