Compactification and distance on Teichmüller space via renormalized volume

Abstract

We introduce a variant of horocompactification which takes “directions” into account. As an application, we construct a compactification of the Teichmüller 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 Teichmüller 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.