On the Hyperbolic Gluing Equations and Representations of Fundamental Groups of Closed 3-Manifolds

TL;DR

This paper investigates solutions to hyperbolic gluing equations for closed 3-manifolds, showing the existence of uncountably many solutions with specific volume and representation properties.

Contribution

It establishes the existence of uncountably many solutions to hyperbolic gluing equations associated with a given fundamental group representation, extending understanding beyond hyperbolic cases.

Findings

Uncountably many solutions to hyperbolic gluing equations exist.

Solutions have representations conjugate to the original.

Volumes of solutions match the original representation's volume.

Abstract

We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many solutions to the hyperbolic gluing equation whose associated representations are conjugate to the given representation, and whose volumes are equal to the volume of the given representation.