Thurston's compactification via geodesic currents: The case of non-compact finite area surfaces

TL;DR

This paper extends Bonahon's construction of Thurston's compactification using geodesic currents from closed surfaces to finite area non-compact surfaces, broadening the applicability of the method.

Contribution

It introduces a variant of Bonahon's approach that works for finite area non-compact surfaces, filling a gap in the existing theory.

Findings

Successfully generalizes Thurston's compactification to finite area surfaces

Provides a new method applicable to non-compact surfaces

Enhances understanding of geodesic currents in broader contexts

Abstract

In [Bon88], Bonahon gave a construction of Thurston's compactification of Teichm{\"u}ller space using geodesic currents. His argument only applies in the case of closed surfaces, and there are good reasons for that. We present a variant which applies to surfaces of finite area.