Horoboundary and rigidity of filling geodesic currents

TL;DR

This paper introduces a new asymmetric metric on the space of projective filling geodesic currents on hyperbolic surfaces, extending Thurston's metric and related structures, and establishes a rigidity result distinguishing different surface genera.

Contribution

It extends the asymmetric metric to broader contexts including Hitchin and Anosov representations, and identifies the horofunction compactification with the space of projective geodesic currents.

Findings

The metric extends Thurston's metric to various representation spaces.

The horofunction compactification matches the space of projective geodesic currents.

Different surface genera yield non-isometric metric spaces.

Abstract

We endow the space of projective filling geodesic currents on a closed hyperbolic surface with a natural asymmetric metric extending Thurston's asymmetric metric on Teichm\"uller space, as well as analogous metrics arising from Hitchin representations. More generally, we show that this metric extends beyond surface groups and geodesic currents, and encompasses metrics associated with Anosov representations of Gromov hyperbolic groups. We identify the horofunction compactification of the space of projective filling currents equipped with this metric with the space of projective geodesic currents. As a consequence, we obtain a rigidity result: the metric spaces of projective filling geodesic currents associated with closed surfaces of distinct genera are not isometric.