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

TL;DR

This paper extends Thurston's boundary concept to infinite-dimensional Teichmüller spaces of noncompact surfaces using geodesic currents, addressing challenges posed by noncompactness and establishing convergence results for earthquake paths.

Contribution

It introduces a new bordification of Teichmüller space for noncompact surfaces utilizing geodesic currents and develops uniformity conditions to handle noncompactness.

Findings

Constructed a Thurston-type boundary for noncompact surfaces.

Established convergence of earthquake paths in the new setting.

Provided technical tools for future research on Teichmüller 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.