Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing Equations

TL;DR

This paper investigates conditions under which solutions to hyperbolic gluing equations imply the essentiality of edges in ideal triangulations of 3-manifolds, extending to generalized equations for cone-manifold structures.

Contribution

It proves that solutions to hyperbolic gluing equations ensure all edges are essential and extends these results to generalized equations for constructing hyperbolic cone-manifolds.

Findings

Solutions imply all edges are essential

Extension to generalized equations for cone-manifold structures

Construction of hyperbolic cone-manifolds with prescribed singularities

Abstract

Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of the hyperbolic gluing equations, which enables the construction of hyperbolic cone-manifold structures on N with singular locus contained in the 1-skeleton of the triangulation.