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Toy Teichmüller spaces of real dimension 2: the pentagon and the punctured triangle

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Abstract

We study two 2-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed number of sides, and their geodesics diverge at most linearly.

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Notes

  1. Masur only considers closed surfaces in [17], but the proof extends essentially verbatim to surfaces with boundary. Alternatively, one can apply Masur’s theorem to the doubles of the surfaces in question. As we noted in Theorem 5, distance is preserved by doubling.

  2. A set is totally geodesic if it contains the bi-infinite geodesic through any two of its points.

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Acknowledgements

This research was conducted during the 2016 Fields Undergraduate Summer Research Program. The authors thank the Fields Institute for providing this opportunity. MFB was partially supported by a postdoctoral research scholarship from the Fonds de recherche du Québec – Nature et technologies.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK

    Yudong Chen

  2. School Mathematics and Computer Sciences, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine

    Roman Chernov

  3. Departamento de Matemáticas, Universidad de Guanajuato, Jalisco s/n Mineral de Valenciana, C.P. 36240, Guanajuato, GTO, Mexico

    Marco Flores

  4. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Maxime Fortier Bourque

  5. Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, 37673, Korea

    Seewoo Lee

  6. Amherst College, 220 South Pleasant Street, Amherst, MA, 01002, USA

    Bowen Yang

Authors
  1. Yudong Chen
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  2. Roman Chernov
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  3. Marco Flores
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  4. Maxime Fortier Bourque
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  5. Seewoo Lee
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  6. Bowen Yang
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Correspondence to Maxime Fortier Bourque.

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Chen, Y., Chernov, R., Flores, M. et al. Toy Teichmüller spaces of real dimension 2: the pentagon and the punctured triangle. Geom Dedicata 197, 193–227 (2018). https://doi.org/10.1007/s10711-018-0325-6

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  • DOI: https://doi.org/10.1007/s10711-018-0325-6

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