Infinitely many virtual geometric triangulations

TL;DR

This paper proves that every cusped hyperbolic 3-manifold has a finite cover with infinitely many geometric ideal triangulations, and this extends to long Dehn fillings, using advanced subgroup separability techniques.

Contribution

It introduces a new conjugacy separability theorem and constructs covers with infinitely many triangulations, advancing understanding of 3-manifold triangulations.

Findings

Existence of finite covers with infinitely many triangulations

Extension of triangulation results to long Dehn fillings

Introduction of a new conjugacy separability theorem

Abstract

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.