Determining hyperbolic 3-manifolds by their surfaces

TL;DR

This paper proves that the set of surface subgroups uniquely determines the commensurability class of a closed, orientable hyperbolic 3-manifold, and that only finitely many such manifolds share the same surface set.

Contribution

It establishes a new link between surface subgroups and the classification of hyperbolic 3-manifolds, showing the set of surfaces determines the manifold's class.

Findings

Surface subgroups determine the commensurability class.

Finitely many manifolds share the same set of surfaces.

The set of surfaces uniquely characterizes the manifold class.

Abstract

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.