Ending lamination theorem

Remarks

Thurston conjectured the theorem. Brock, Canary and Minsky proved it, first for manifolds homeomorphic to $S \times \R$ for a surface $S$, building on earlier work of Minsky. The general case uses the tameness theorem, which makes the ends of $M$ products of a surface and a half-line, so that end invariants make sense. The proof builds a combinatorial model manifold from the end invariants, using the curve complex, and shows that $M$ is bilipschitz to this model. For example, a quasi-Fuchsian manifold $S \times \R$ has two geometrically finite ends. The theorem then recovers Bers simultaneous uniformization: the manifold is determined by a pair of points in the Teichmüller space of $S$.

Going deeper

For a student

The theorem completes the classification of hyperbolic $3$-manifolds with finitely generated fundamental group. Together with tameness, it was used to prove the density theorem: every finitely generated Kleinian group is a limit of geometrically finite ones. It also makes rigorous Thurston's picture of doubly degenerate surface groups, such as the fiber group of a fibered hyperbolic $3$-manifold.