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THE SIMPLE SEPARATING SYSTOLE FOR HYPERBOLIC SURFACES OF LARGE GENUS

Part of: Low-dimensional topology Deformations of analytic structures

Published online by Cambridge University Press:  31 May 2021

Hugo Parlier ,
Yunhui Wu  and
Yuhao Xue Open the ORCID record for Yuhao Xue [Opens in a new window]
Hugo Parlier
Affiliation:
Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg (hugo.parlier@uni.lu)
Yunhui Wu
Affiliation:
Yau Mathematical Sciences Center & Department of Mathematical Sciences, Tsinghua University, Beijing, China (yunhui_wu@tsinghua.edu.cn); (xueyh18@mails.tsinghua.edu.cn)
Yuhao Xue
Affiliation:
Yau Mathematical Sciences Center & Department of Mathematical Sciences, Tsinghua University, Beijing, China (yunhui_wu@tsinghua.edu.cn); (xueyh18@mails.tsinghua.edu.cn)
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Abstract

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil–Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

Keywords

Random surfacesSimple separating systoleWeil-Petersson volumes

MSC classification

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory
Secondary: 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2205 - 2214
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Brooks, R., Some relations between spectral geometry and number theory, in Topology’90 (Columbus, OH, 1990), Ohio State University Mathematical Research Institute Book Series, volume 1, pp. 61–75 (de Gruyter, Berlin, 1992).Google Scholar
Brooks, R. and Makover, E., Random construction of Riemann surfaces, J. Differential Geom. 68(1) (2004), 121–157.CrossRefGoogle Scholar
Buser, P., Geometry and Spectra of Compact Riemann Surfaces, Modern Birkhäuser Classics (Birkhäuser Boston, Inc., Boston, MA, 2010). Reprint of the 1992 edition.CrossRefGoogle Scholar
Gromov, M., Filling Riemannian manifolds, J. Differential Geom. 18(1) (1983), 1–147.CrossRefGoogle Scholar
Guth, L., Parlier, H. and Young, R., Pants decompositions of random surfaces, Geom. Funct. Anal. 21(5) (2011), 1069–1090.CrossRefGoogle Scholar
Mirzakhani, M., Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom. 94(2) (2013), 267–300.CrossRefGoogle Scholar
Mirzakhani, M. and Petri, B., Lengths of closed geodesics on random surfaces of large genus, Comment. Math. Helv. 94(4) (2019), 869–889.CrossRefGoogle Scholar
Nie, X., Wu, Y. and Xue, Y., ‘Large genus asymptotics for lengths of separating closed geodesics on random surfaces’, Preprint, 2020, https://ui.adsabs.harvard.edu/abs/2020arXiv200907538N.Google Scholar
Petri, B., Random regular graphs and the systole of a random surface, J. Topol. 10(1) (2017), 211–267.CrossRefGoogle Scholar
Petri, B. and Thäle, C., Poisson approximation of the length spectrum of random surfaces, Indiana Univ. Math. J. 67(3) (2018), 1115–1141.CrossRefGoogle Scholar
Sabourau, S., Asymptotic bounds for separating systoles on surfaces, Comment. Math. Helv. 83(1) (2008), 35–54.CrossRefGoogle Scholar
Wolpert, S., The Fenchel-Nielsen deformation, Ann. of Math. (2) 115(3) (1982), 501–528.CrossRefGoogle Scholar