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

Hidetoshi Masai

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arXiv · 2108.06059

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

Hidetoshi Masai

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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.

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Hidetoshi Masai. 2021-08-13. Compactification and distance on Teichm\"uller space via renormalized volume. https://arxiv.org/abs/2108.06059

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