Hyperbolic limits of Cantor set complements in the sphere

TL;DR

This paper investigates how certain hyperbolic 3-manifolds can be approximated by complements of Cantor sets in the 3-sphere, revealing new geometric limits and embedding properties.

Contribution

It establishes the existence of Cantor set complements in the 3-sphere that approximate hyperbolic 3-manifolds with specific topological conditions.

Findings

Existence of hyperbolic Cantor set complements approximating given manifolds

Construction of geometric limits of these complements

Insights into embedding properties in the 3-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 $N_n\rightarrow M$ geometrically.