Thurston's boundary to infinite-dimensional Teichmüller spaces: geodesic currents

Abstract

Let $X_0$ be a complete borderless infinite area hyperbolic surface. We introduce Thurston's boundary to the Teichmüller space $T(X_0)$ of the surface $X_0$ using Liouville (geodesic) currents. Thurston's boundary to $T(X_0)$ is identified with the space $PM{L}_{bdd}(X_0)$ of projective bounded measured laminations on $X_0$ which naturally extends Thurston's result for closed surfaces. Moreover, the quasiconformal mapping class group $MC{G}_{qc}(X_0)$ acts continuously on the closure $T(X_0)\cup PM{L}_{bdd}(X_0)$.

24 pages, 5 figures

Cited by in corpus (1)

• A Thurston boundary for infinite-dimensional Teichmüller spaces