Compactification and distance on Teichm\"uller space via renormalized volume

TL;DR

This paper introduces a new compactification of Teichmüller space using renormalized volume, enabling a novel distance where pseudo-Anosov translation lengths relate directly to hyperbolic volume.

Contribution

It constructs a compactification of Teichmüller space via renormalized volume and defines a new distance where pseudo-Anosov translation lengths equal hyperbolic volume.

Findings

New compactification of Teichmüller space using renormalized volume

A novel distance on Teichmüller space related to hyperbolic volume

Connection between pseudo-Anosov translation length and hyperbolic volume

Abstract

We introduce a variant of horocompactification which takes "directions" into account. As an application, we construct a compactification of the Teichm\"uller spaces via the renormalized volume of quasi-Fuchsian manifolds. Although we observe that the renormalized volume itself does not give a distance, the compactification allows us to define a new distance on the Teichm\"uller space. We show that the translation length of pseudo-Anosov mapping classes with respect to this new distance is precisely the hyperbolic volume of their mapping tori. A similar compactification via the Weil-Petersson metric is also discussed.