Determining hyperbolic 3-manifold groups by their finite quotients

Abstract

It is conjectured that if $M$ and $N$ are finite volume hyperbolic 3-manifolds, then $M$ and $N$ are isometric if and only if their fundamental groups have the same finite quotients. The most general case in which the conjecture is known to hold is when $M$ is a punctured torus bundle over the circle, by work of Bridson, Reid and Wilton. Distinguishing a single pair of hyperbolic 3-manifold groups by naively enumerating finite quotients with a computer can take days. In this talk, I will describe the relatively non-naive computational verification that the conjecture holds when both $M$ and $N$ are chosen from the ~70,000 census manifolds included in SnapPy, and the theory behind it.