Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

TL;DR

This paper discusses an algorithm derived from Thurston's hyperbolization theorem and normal surface theory to determine if a compact orientable 3-manifold with torus boundary admits a hyperbolic metric, aiding in resolving a conjecture.

Contribution

It presents a concrete algorithm for hyperbolicity detection in 3-manifolds with torus boundary, connecting theoretical conjectures with computational methods.

Findings

Algorithm successfully determines hyperbolicity for given 3-manifolds.

Supports the conjecture of Gabai, Meyerhoff, and Milley through computational verification.

Bridges theoretical topology and practical computation in 3-manifold geometry.

Abstract

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.