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.

36 pages, 2 figures, v3: minor edit. v2:minor edit, fixed typos and notations, added some explanations

References in corpus (2)

• On the renormalized volume of hyperbolic 3-manifolds
• The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores