An Asymmetric Thurston Metric on Currents and Its Horoboundary

The classical asymmetric Thurston metric measures the minimal Lipschitz distortion between two hyperbolic metrics on a surface. In this talk, I will present a broad generalization of this metric, encompassing several recently introduced asymmetric metrics arising in the context of Anosov representations. This construction, in particular, extends the Thurston metric to the space of projective filling geodesic currents, and we study its associated horoboundary. As an application, we show that two surfaces have non-isometric spaces of filling currents if and only if they have different genera. This is joint work with Meenakshy Jyothis.