Stable Teichmüller quasigeodesics and ending laminations

Abstract

We characterize which cobounded quasigeodesics in the Teichmüller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path γ in T, we show that γ is a quasigeodesic with finite Hausdor distance from some geodesic if and only if the canonical hyperbolic plane bundle over γ is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky's proof of Thurston's ending lamination conjecture for such manifolds.