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Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

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Abstract

We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is \(\le 4{\text {arcsinh}}(1)\), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

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Notes

  1. There are, in fact, uncountably many leaves in \(\lambda \), and only finitely many lie on the boundary.

  2. For the Thurston boundary of Teichmüller spaces of surfaces with boundary, see [2].

  3. This measured lamination is unique when \(S=S_{1,1}\).

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Acknowledgements

We thank Kasra Rafi and David Dumas for correspondence and for his interest in our work, François Guéritaud for his very thoughtful and enlightening responses to our many questions and Vincent Alberge who read a preliminary version of this paper. We also thank the referee whose corrections and suggestions substantially improved this paper.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Office 261, Jinchunyuan West Bldg., Haidian District, Beijing, 100084, China

    Yi Huang

  2. Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Athanase Papadopoulos

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  1. Yi Huang
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  2. Athanase Papadopoulos
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Huang, Y., Papadopoulos, A. Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space. Geom Dedicata 214, 465–488 (2021). https://doi.org/10.1007/s10711-021-00624-z

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