Geodesic currents and Teichmüller space

Abstract

Consider a hyperbolic surface X of infinite area. The Liouville map L assigns to any quasiconformal deformation of X a measure on the space G( X ̃ ) of geodesics of the universal covering X̃ of X. We show that the Liouville map L is a homeomorphism from the Teichmüller space T(X) onto its image, and that the image L( T(X)) is closed and unbounded. The set of asymptotic rays to L( T(X)) consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for T(X) . The action of the quasiconformal mapping class group on T(X) continuously extends to this boundary for T(X) .