Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem

TL;DR

This paper provides a complete, elementary proof of Thurston's hyperbolic Dehn filling theorem using negatively oriented ideal triangulations, avoiding assumptions of existing ideal triangulations and smoothness conditions.

Contribution

It introduces a novel approach employing negatively oriented tetrahedra and a subdivision of Epstein-Penner decompositions, broadening the understanding of hyperbolic Dehn fillings.

Findings

Complete proof of Thurston's hyperbolic Dehn filling theorem.

Elementary and self-contained analysis of Dehn filling coefficients.

Avoids assumptions of smoothness and existence of genuine ideal triangulations.

Abstract

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. This forces us to deal with negatively oriented tetrahedra. Our analysis of the set of hyperbolic Dehn filling coefficients is elementary and self-contained. In particular, it does not assume smoothness of the complete point in the variety of deformations.