An extension of the Thurston metric to projective filling currents

We study the geometry of the space of projectivized filling geodesic currents \(\mathbb {P}\mathcal {C}_{fill}(S)\). Bonahon showed that Teichmüller space, \(\mathcal {T}(S)\) embeds into \(\mathbb {P}\mathcal {C}_{fill}(S)\). We extend the symmetrized Thurston metric from \(\mathcal {T}(S)\) to the entire (projectivized) space of filling currents, and we show that \(\mathcal {T}(S)\) is isometrically embedded into the bigger space. Moreover, we show that there is no quasi-isometric projection back down to \(\mathcal {T}(S)\). Lastly, we study the geometry of a length-minimizing projection from \(\mathbb {P}\mathcal {C}_{fill}(S)\) to \(\mathcal {T}(S)\) defined previously by Hensel and the author.