A Thurston boundary for infinite-dimensional Teichm\"uller spaces

Abstract

For a compact surface $X_0$, Thurston introduced a compactification of its Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichm\"uller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$, may be of independent interest.