Proving a manifold to be hyperbolic once it has been approximated to be so

TL;DR

This paper introduces a rigorous method to confirm that a 3-manifold has a complete hyperbolic structure by leveraging approximations from computational tools and applying Kantorovich's theorem for proof.

Contribution

It develops a novel approach combining triangulation, Newton's method, and Kantorovich's theorem to conclusively prove hyperbolicity of manifolds approximated by SnapPea.

Findings

Successfully proves hyperbolicity for all manifolds in the SnapPea cusped census.

Provides a method to convert approximate solutions into rigorous proofs of hyperbolic structure.

Enhances the reliability of computational topology tools for 3-manifold analysis.

Abstract

The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximations of SNAP, a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating the manifold, identifying consistency and completeness equations with respect to this triangulation, and then trying to solve the system of equations using Newton's Method. This produces an approximate, not actual solution. The method developed here uses Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure.