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The shortest non-simple closed geodesics on hyperbolic surfaces

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Abstract

This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large k, the shortest closed geodesics with at least k self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length \(2{\,\mathrm arccosh}(2k+1)\).

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Acknowledgements

This paper was finished while the first author was a semester long visitor at the University of Michigan. It is a pleasure to thank the University of Michigan and Richard Canary for their support. The authors thank a variety of people they have discussed these ideas with including Binbin Xu and Peter Buser.

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Authors and Affiliations

  1. The Graduate Center, City University of New York, New York, NY, 10016, USA

    Ara Basmajian

  2. Hunter College, City University of New York, 695 Park Ave, New York, NY, 10065, USA

    Ara Basmajian

  3. DMATH, FSTM, University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Hugo Parlier

  4. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA

    Hanh Vo

Authors
  1. Ara Basmajian
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  2. Hugo Parlier
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  3. Hanh Vo
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Correspondence to Ara Basmajian.

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Supported by a Grant from the Simons foundation (359956, A.B.) and PSC-CUNY Award 65245-00 53.

Supported by the Luxembourg National Research Fund OPEN Grant O19/13865598.

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Basmajian, A., Parlier, H. & Vo, H. The shortest non-simple closed geodesics on hyperbolic surfaces. Math. Z. 306, 8 (2024). https://doi.org/10.1007/s00209-023-03401-8

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