An extension of the Thurston metric to projective filling currents

TL;DR

This paper extends the Thurston metric to the space of projective filling currents, embedding Teichmüller space isometrically and analyzing the geometry of projections between these spaces.

Contribution

It generalizes the Thurston metric to the space of filling currents and studies the geometric properties of embeddings and projections involving Teichmüller space.

Findings

Teichmüller space embeds isometrically into the space of filling currents.

No quasi-isometric projection exists from the filling currents space to Teichmüller space.

The geometry of length-minimizing projections is characterized.

Abstract

We study the geometry of the space of projectivized filling geodesic currents $\mathbb P \mathcal C_{fill}(S)$. Bonahon showed that Teichm\"uller 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.