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Short closed geodesics with self-intersections

Part of: Deformations of analytic structures Riemann surfaces Global differential geometry

Published online by Cambridge University Press:  24 January 2020

VIVEKA ERLANDSSON  and
HUGO PARLIER
VIVEKA ERLANDSSON
Affiliation:
School of Mathematics, University of Bristol, e-mail: [email protected]
HUGO PARLIER
Affiliation:
Department of Mathematics, University of Luxembourg, e-mail: [email protected]
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Abstract

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

MSC classification

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory
Secondary: 30F10: Compact Riemann surfaces and uniformization 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 53C22: Geodesics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 623 - 638
Copyright
© Cambridge Philosophical Society 2020

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References

REFERENCES

Aougab, T., Gaster, J., Patel, P. and Sapir, J.. Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett. 24(3) (2017), 593–617.Google Scholar
Balacheff, F. and Parlier, H.. Bers’ constants for punctured spheres and hyperelliptic surfaces. J. Topol. Anal. 4(3) (2012), 271–296.CrossRefGoogle Scholar
Basmajian, A.. The stable neighborhood theorem and lengths of closed geodesics. Proc. Amer. Math. Soc. 119(1) (1993), 217–224.CrossRefGoogle Scholar
Basmajian, A.. Universal length bounds for non-simple closed geodesics on hyperbolic surfaces. J. Topol. 6(2) (2013), 513–524.CrossRefGoogle Scholar
Basmajian, A.. Short geodesics on a hyperbolic surface. In Recent advances in mathematics, Ramanujan Math. Soc. Lect. Notes Ser. vol. 21, pages 39–43 (Ramanujan Math. Soc., Mysore, 2015).Google Scholar
Basmajian, A. and Kim, Y.. Geometrically infinite surfaces with discrete length spectra. Geom. Dedicata 137 (2008), 219–240.CrossRefGoogle Scholar
Bers, L.. An inequality for Riemann surfaces. In Differential Geometry and Complex Analysis (Springer, Berlin, 1985), Pages 87–93.CrossRefGoogle Scholar
Buser, P.. Geometry and spectra of compact Riemann surfaces. Progr. Math. vol. 106. (Birkhäuser Boston, Inc., Boston, MA, 1992).Google Scholar
Chas, M.. Relations between word length, hyperbolic length and self-intersection number of curves on surfaces. In Recent advances in mathematics. Ramanujan Math. Soc. Lect. Notes Ser. vol. 21, pages 45–75 (Ramanujan Math. Soc., Mysore, 2015).Google Scholar
Chas, M. and Phillips, A.. Self-intersection numbers of curves on the punctured torus. Experiment. Math. 19(2) (2010), 129–148.Google Scholar
Erlandsson, V. and Souto, J.. Counting curves in hyperbolic surfaces. Geom. Funct. Anal. 26(3) (2016), 729–777.Google Scholar
Parlier, H.. A short note on short pants. Canad. Math. Bull. 57(4) (2014), 870–876.CrossRefGoogle Scholar
Sapir, J.. Bounds on the number of non-simple closed geodesics on a surface. Int. Math. Res. Not. IMRN, 2016(24) (2020), 7499–7545.CrossRefGoogle Scholar
Sapir, J.. Lower bound on the number of non-simple closed geodesics on surfaces. Geom. Dedicata, 184(1) (2020), 1–25.CrossRefGoogle Scholar