Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-30T15:10:59.054Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2205 - 2214
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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