Three dimensional manifolds, Kleinian groups and hyperbolic geometry

TLDR

The uniformization theory for Riemann surfaces, extended to hyperbolic structures for surfaces of negative Euler characteristic, inspired a search for an analogous theory in three dimensions, where 3‑manifolds are far more complex and until recently no comparable framework existed. The situation has changed, allowing a new conjectural picture of 3‑manifolds to be proposed. 1.

Abstract

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the