Geodesics and compression bodies

TL;DR

This paper investigates the geometric properties of hyperbolic structures on a specific type of 3-manifold called a compression body, focusing on the core tunnel's isotopy to a geodesic and providing computational tools for visualization.

Contribution

It proposes a conjecture relating the core tunnel to geodesics in hyperbolic structures and offers algorithms and implementations for computing and visualizing Ford domains.

Findings

The conjecture holds for many geometrically finite structures.

Developed an algorithm to compute Ford domains.

Created a visualization tool supporting the conjecture.

Abstract

We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of C. We conjecture that, in any geometrically finite structure on C, the core tunnel is isotopic to a geodesic. By considering Ford domains, we show this conjecture holds for many geometrically finite structures. Additionally, we give an algorithm to compute the Ford domain of such a manifold, and a procedure which has been implemented to visualize many of these Ford domains. Our computer implementation gives further evidence for the conjecture.