Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds

Abstract

We study two boundaries for the Teichmüller space of a surface Teich(S) due to L. Bers and W. Thurston. Each point in Bers's boundary is a hyperbolic 3-manifold with an associated geodesic lamination on S, its end-invariant, while each point in Thurston's is a measured geodesic lamination, up to scale. When dim_ℂ(Teich(S))>1, we show that the end-invariant is not a continuous map to Thurston's boundary modulo forgetting the measure with the quotient topology. We recover continuity by allowing as limits maximal measurable sublaminations of Hausdorff limits and enlargements thereof.