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.