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.