Stable Teichmüller quasigeodesics and ending laminations

Abstract

We characterize which cobounded quasigeodesics in the Teichmüller space $\mathcal{T}$ of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path $\gamma$ in $\mathcal{T}$, we show that $\gamma$ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over $\gamma$ 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.