An extension of the Thurston metric to projective filling currents

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