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Lower bound on the number of non-simple closed geodesics on surfaces

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Abstract

We give a lower bound on the number of non-simple closed geodesics on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number \(K = K(L)\) goes from a constant to a quadratic function of L, the number of closed geodesics transitions from polynomial to exponential in L. We show upper bounds on the number of such geodesics in a subsequent paper.

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Acknowledgments

This paper is part of the author’s Ph.D. thesis, which was completed under her advisor, Maryam Mirzakhani. The author would especially like to thank her for the many conversations that led to this work. The author would also like to thank Jayadev Athreya, Steve Kerckhoff and Chris Leininger for their help and support.

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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 Green St, Urbana, IL, 61801, USA

    Jenya Sapir

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  1. Jenya Sapir
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Correspondence to Jenya Sapir.

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Sapir, J. Lower bound on the number of non-simple closed geodesics on surfaces. Geom Dedicata 184, 1–25 (2016). https://doi.org/10.1007/s10711-016-0155-3

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  • DOI: https://doi.org/10.1007/s10711-016-0155-3

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