Skip to main content
Log in

Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\epsilon \)

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Let \(\mathcal {M}_g\) be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Petersson metric. In this paper, we show that for any \(\epsilon >0\), as genus g goes to infinity, a generic surface \(X\in \mathcal {M}_g\) satisfies that the first eigenvalue \(\lambda _1(X)>\frac{3}{16}-\epsilon \). As an application, we also show that a generic surface \(X\in \mathcal {M}_g\) satisfies that the diameter \({{\,\mathrm{diam}\,}}(X)<(4+\epsilon )\ln (g)\) for large genus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. P. Buser, M. Burger and J. Dodziuk. Riemann surfaces of large genus and large\(\lambda _1\), Geometry and analysis on manifolds (Katata/Kyoto, 1987), Lecture Notes in Math., vol. 1339, Springer, Berlin (1988), pp. 54–63.

  2. N. Bergeron. The spectrum of hyperbolic surfaces, Universitext, Springer, Cham; EDP Sciences, Les Ulis, 2016, Appendix C by Valentin Blomer and Farrell Brumley, Translated from the 2011 French original by Brumley [2857626].

  3. R. Brooks and E. Makover. Random construction of Riemann surfaces, J. Differ. Geom., (1)68 (2004), 121–157.

    Article  MathSciNet  Google Scholar 

  4. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, vol. 106. Birkhäuser Boston, Inc., Boston, MA (1992).

    MATH  Google Scholar 

  5. S.Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Math. Z., (3)143 (1975), 289–297.

    Article  MathSciNet  Google Scholar 

  6. V. Delecroix, E. Goujard, P. Zograf and A. Zorich. Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves. arXiv e-prints (2020). arXiv:2007.04740.

  7. S. Gelbart and H. Jacquet. A relation between automorphic representations of \({\rm GL}(2)\)and\({\rm GL}(3)\). Ann. Sci. École Norm. Sup. (4), (4)11 (1978), 471–542.

  8. C. Gilmore, E. Le Masson, T. Sahlsten and J. Thomas. Short geodesic loops and \(L^p\) norms of eigenfunctions on large genus random surfaces. Geom. Funct. Anal., (1)31 (2021), 62–110.

  9. L. Guth, H. Parlier and R. Young. Pants decompositions of random surfaces. Geom. Funct. Anal., (5)21 (2011), 1069–1090.

    Article  MathSciNet  Google Scholar 

  10. D.A. Hejhal. The Selberg Trace Formula for\({\rm PSL}(2,R)\). Vol. I, Lecture Notes in Mathematics, Vol. 548. Springer, Berlin-New York (1976).

  11. W. Hide. Spectral gap for Weil-Petersson random surfaces with cusps. arXiv e-prints (2021. arXiv:2107.14555.

  12. W. Hide and M. Magee. Near optimal spectral gaps for hyperbolic surfaces. arXiv e-prints (2021). arXiv:2107.05292.

  13. H. Huber. Über den ersten Eigenwert des Laplace-Operators auf kompakten Riemannschen Flächen. Comment. Math. Helv., 49 (1974), 251–259.

    Article  MathSciNet  Google Scholar 

  14. H. Iwaniec. Selberg’s lower bound of the first eigenvalue for congruence groups, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA (1989), pp. 371–375.

  15. H. Iwaniec, The lowest eigenvalue for congruence groups, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA (1996), pp. 203–212.

  16. H. Iwaniec, Spectral methods of automorphic forms, second ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid (2002).

  17. F. Jenni. Über den ersten Eigenwert des Laplace-Operators auf ausgewählten Beispielen kompakter Riemannscher Flächen. Comment. Math. Helv., (2)59 (1984), 193–203.

    Article  MathSciNet  Google Scholar 

  18. H.H. Kim. Functoriality for the exterior square of \({\rm GL}_4\) and the symmetric fourth of \({\rm GL}_2\). J. Amer. Math. Soc., (1)16 (2003), 139–183, With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.

  19. H.H. Kim and F. Shahidi. Functorial products for \({\rm GL}_2\times {\rm GL}_3\) and the symmetric cube for \({\rm GL}_2\). Ann. of Math. (2), (3)155 (2002), 837–893, With an appendix by Colin J. Bushnell and Guy Henniart.

  20. S.P. Lalley. Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math., (1-2)163 (1989), 1–55.

    Article  MathSciNet  Google Scholar 

  21. W. Luo, Z. Rudnick and P. Sarnak. On Selberg’s eigenvalue conjecture. Geom. Funct. Anal., (2)5 (1995), 387–401.

    Article  MathSciNet  Google Scholar 

  22. M. Lipnowski and A. Wright. Towards optimal spectral gaps in large genus. arXiv e-prints (2021). arXiv:2103.07496.

  23. M. Magee. Letter to Bram Petri. https://www.maths.dur.ac.uk/users/michael.r.magee/diameter.pdf (2020).

  24. M. Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math., (1)167 (2007), 179–222.

    Article  MathSciNet  Google Scholar 

  25. M. Mirzakhani. Weil-Petersson volumes and intersection theory on the moduli space of curves. J. Am. Math. Soc., (1)20 (2007), 1–23.

    Article  MathSciNet  Google Scholar 

  26. M. Mirzakhani. On Weil-Petersson volumes and geometry of random hyperbolic surfaces, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi (2010), pp. 1126–1145.

  27. M. Mirzakhani. Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus. J. Differ. Geom., (2)94 (2013), 267–300.

    Article  MathSciNet  Google Scholar 

  28. M. Magee, F. Naud and D. Puder. A random cover of a compact hyperbolic surface has relative spectral gap\(\frac{3}{16}-\epsilon \). arXiv e-prints (2020). arXiv:2003.10911.

  29. S. Mondal. On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces. Math. Z., (1-2)281 (2015), 333–348.

    Article  MathSciNet  Google Scholar 

  30. L. Monk. Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus. Anal. PDE (2020), to appear.

  31. M. Mirzakhani and B. Petri. Lengths of closed geodesics on random surfaces of large genus. Comment. Math. Helv., (4)94 (2019), 869–889.

    Article  MathSciNet  Google Scholar 

  32. M. Mirzakhani and P. Zograf. Towards large genus asymptotics of intersection numbers on moduli spaces of curves. Geom. Funct. Anal., (4)25 (2015), 1258–1289.

    Article  MathSciNet  Google Scholar 

  33. X. Nie, Y. Wu and Y. Xue. Large genus asymptotics for lengths of separating closed geodesics on random surfaces. arXiv e-prints (2020). arXiv:2009.07538.

  34. H. Parlier, Y. Wu and Y. Xue. The simple separating systole for hyperbolic surfaces of large genus. J. Inst. Math. Jussieu (2021), to appear.

  35. P. Sarnak. Selberg’s eigenvalue conjecture. Notices Am. Math. Soc., (11)42 (1995), 1272–1277.

    MathSciNet  MATH  Google Scholar 

  36. A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.), 20 (1956), 47–87.

    MathSciNet  MATH  Google Scholar 

  37. A. Selberg. On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I. (1965), pp. 1–15.

  38. S. Wolpert. The Fenchel-Nielsen deformation. Ann. of Math. (2), (3)115 (1982), 501–528.

    Article  MathSciNet  Google Scholar 

  39. S.A. Wolpert. Families of Riemann surfaces and Weil-Petersson geometry, CBMS Regional Conference Series in Mathematics, vol. 113, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2010).

  40. A. Wright. A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces. Bull. Amer. Math. Soc. (N.S.), (3)57 (2020), 359–408.

    Article  MathSciNet  Google Scholar 

  41. Y. Wu and Y. Xue. Small eigenvalues of closed Riemann surfaces for large genus. Trans. Am. Math. Soc. (2021), to appear.

Download references

Acknowledgements

The authors would like to thank Yang Shen for helpful discussions on the key counting result Theorem 4. They also would like to thank Prof. S. T. Yau for his interests on this work. Both authors are supported by the NSFC Grant No. 12171263, and the first named author is also partially supported by a grant from Tsinghua University. We are also grateful to the referees for helpful comments and suggestions which improve this article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yunhui Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Y., Xue, Y. Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\epsilon \). Geom. Funct. Anal. 32, 340–410 (2022). https://doi.org/10.1007/s00039-022-00595-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-022-00595-7

Mathematics Subject Classification

Navigation