Hyperbolic Handlebody Complements in 3-Manifolds

TL;DR

This paper proves that certain 3-manifolds contain handlebodies of arbitrary genus with hyperbolic complements and provides volume bounds for some of these handlebody complements using an extended octahedral decomposition.

Contribution

It introduces a method to find handlebodies of arbitrary genus with hyperbolic complements in 3-manifolds and extends octahedral decomposition techniques for volume estimation.

Findings

Existence of handlebodies of arbitrary genus with hyperbolic complements

Extension of octahedral decomposition for volume bounds

New bounds on volume for specific handlebody complements

Abstract

Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.