On the Thurston boundary and the relatively hyperbolic boundary of Teichmüller space

Abstract

We obtain a result about the relation between the Thurston boundary and the relatively hyperbolic boundary of Teichmüller space. Precisely, we prove that the identity map on Teichmüller space extends to a continuous surjective map from the subset of the Thurston boundary consisting of minimal measured foliations to the relatively hyperbolic boundary. As an application, to relate the Thurston compactification and the Teichmüller compactification of Teichmüller space, we construct a new compactification of Teichmüller space which is weaker than the Thurston compactification and the Teichmüller compactification.