Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds

TL;DR

This paper demonstrates that hyperbolic structures on closed 3-manifolds can be constructed from solutions to gluing equations via Thurston's spinning construction, leading to a new algorithm for detecting hyperbolic structures.

Contribution

It introduces a novel approach combining Thurston's spinning construction with volume rigidity to construct hyperbolic structures from gluing equations.

Findings

Hyperbolic structures can be derived from solutions to gluing equations.

A new algorithm for detecting hyperbolic structures on closed 3-manifolds.

Validation of the approach using essential edge triangulations.

Abstract

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures on closed 3-manifolds.