Profinite Rigidity in the SnapPea Census

Abstract

A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72,942 finite volume hyperbolic 3-manifolds have the same finite quotients.

Keywords:

• profinite rigidity
• hyperbolic 3-manifolds
• profinite completions
• computation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E18
• 57M27
• 20E26
• 20-04

Acknowledgments

I thank Martin Bridson for guidance and support while supervising this project as part of my thesis, Alan Reid for a stimulating minicourse at YGGT VI in Oxford (March 2017) which lead to me undertaking this project, and Nathan Dunfield, Jim Howie, Marc Lackenby, Nikolay Nikolov, and Gareth Wilkes for their comments.