Uniform models for hyperbolic 3-manifolds

I'll discuss the problem of predicting the internal structure of a 3-manifold from its end invariants. While this has a pretty satisfying answer for manifolds homeomorphic to (surface)x(interval), we do not have a complete answer in general which is --uniform over an entire deformation space--. In joint work with Ken Bromberg and Dick Canary, we construct such a model for manifolds with incompressible boundary, prove it gives estimates in one direction, and use this to establish an old theorem of Thurston whose original proof is not known.