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Weil-Petersson geometry of Teichmüller–Coxeter complex and its finite rank property

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Abstract

On a Teichmüller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.

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Authors and Affiliations

  1. Tohoku University, Sendai, Japan

    Sumio Yamada

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  1. Sumio Yamada
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Correspondence to Sumio Yamada.

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Yamada, S. Weil-Petersson geometry of Teichmüller–Coxeter complex and its finite rank property. Geom Dedicata 145, 43–63 (2010). https://doi.org/10.1007/s10711-009-9401-2

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  • DOI: https://doi.org/10.1007/s10711-009-9401-2

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