Profinite rigidity in the SnapPea census

TL;DR

This paper provides computational evidence supporting the conjecture that finite volume hyperbolic 3-manifolds are distinguished by their finite quotients, showing no two manifolds in the SnapPea census share the same finite quotients.

Contribution

It offers the first large-scale computational verification that all manifolds in the SnapPea census are distinguished by their finite quotients.

Findings

No two manifolds in the census share the same finite quotients

Supports the conjecture for a broad class of hyperbolic 3-manifolds

Provides a foundation for future theoretical proofs

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.