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.
Funding Statement
The work of the author is partially supported by JSPS KAKENHI Grant Number 19K14525 and 23K03085.
Citation
Hidetoshi MASAI. "Compactification and distance on Teichmüller space via renormalized volume." J. Math. Soc. Japan 76 (3) 673 - 712, July, 2024. https://doi.org/10.2969/jmsj/90429042
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