2021/08/13 by Masai, Hidetoshi · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2108.06059
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.