Hyperbolic limits of Cantor set complements in the sphere

Abstract

Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in ${\mathbb{S}}^3$. Then, if $M$ admits an exhaustion by ${\pi }_1$-injective sub-manifolds there exists cantor sets $C_n\subset {\mathbb{S}}^3$ such that $N_n={\mathbb{S}}^3\setminus C_n$ is hyperbolic and $N_n\to M$ geometrically.

Version accepted for publication at Bulletin of the London Mathematical Society