Coarse models for hyperbolic 3-manifolds

Brian H. Bowditch

We give a characteristion of models of hyperbolic 3-manifolds in terms of Gromov hyperbolicity. Let M be a riemannian manifold with a homotopically equivalence to a closed surface, Sigma, which associates cusps of M to boundary components of Sigma. Under some assumption on M, notably that that universal cover of M is Gromov hyperbolic, we show that the ends of M can be classified as geometrically finite or simply degenerate. To any simply degenerate end we assoicated an ending lamination. We show that any two doubly degenerate manifolds of this sort with the same ending laminations are equivalent, in the sense that their universal covers are equivariantly quasi-isometric. In particular, this applies to constant curvature hyperbolic 3-manifold. A consequence is the ending lamintion conjecture for such manifolds, as proven by Minsky, Brock and Canary.

September 2010.

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