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A Thurston boundary for infinite-dimensional Teichmüller spaces

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Abstract

For a compact surface \(X_0\), Thurston introduced a compactification of its Teichmüller space \({\mathcal {T}}(X_0)\) by completing it with a boundary \(\mathcal {PML}(X_0)\) consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space \({\mathcal {T}}(X_0)\) of a noncompact Riemann surface \(X_0\), using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in \({\mathcal {T}}(X_0)\), may be of independent interest.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, CA, 90089-2532, USA

    Francis Bonahon

  2. Department of Mathematics, Queens College of the City University of New York, 65-30 Kissena Blvd., Flushing, NY, 11367, USA

    Dragomir Šarić

  3. Mathematics Ph.D. Program, Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    Dragomir Šarić

Authors
  1. Francis Bonahon
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  2. Dragomir Šarić
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Correspondence to Dragomir Šarić.

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Communicated by F.C. Marques.

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This research was partially supported by the grants DMS-1102440, DMS-1406559 and DMS-1711297 from the US National Science Foundation, and by the collaboration grant 346391 from the Simons Foundation.

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Bonahon, F., Šarić, D. A Thurston boundary for infinite-dimensional Teichmüller spaces. Math. Ann. 380, 1119–1167 (2021). https://doi.org/10.1007/s00208-021-02148-z

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