Thurston's boundary to infinite-dimensional Teichm\"uller spaces: geodesic currents

TL;DR

This paper extends Thurston's boundary concept to infinite-area hyperbolic surfaces using geodesic currents, identifying it with bounded measured laminations and analyzing the action of the quasiconformal mapping class group.

Contribution

It introduces a Thurston boundary for infinite-area hyperbolic surfaces via Liouville currents, extending Thurston's classical results to new infinite-dimensional settings.

Findings

Thurston's boundary is identified with the space of projective bounded measured laminations.

The boundary is naturally extended from closed to infinite-area surfaces.

The quasiconformal mapping class group acts continuously on the extended space.

Abstract

Let $X_0$ be a complete borderless infinite area hyperbolic surface. We introduce Thurston's boundary to the Teichm\"uller 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 $PML_{bdd}(X_0)$ of projective bounded measured laminations on $X_0$ which naturally extends Thurston's result for closed surfaces. Moreover, the quasiconformal mapping class group $MCG_{qc}(X_0)$ acts continuously on the closure $T(X_0)\cup PML_{bdd}(X_0)$.