A Thurston boundary for infinite-dimensional Teichmüller spaces

Abstract

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal{T}(X_0)$ by completing it with a boundary $\mathcal{P}\mathcal{ML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller 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.

42 pages, 3 figures. Version 2: Minor revisions prior to publication; to appear in Mathematische Annalen

References in corpus (3)

• Limits of Teichmüller geodesics in the Universal Teichmüller space
• Visual sphere and Thurston's boundary of the Universal Teichmüller space
• Thurston's boundary to infinite-dimensional Teichmüller spaces: geodesic currents

Cited by in corpus (2)

• From curves to currents
• On complex extension of the Liouville map