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Large Genus Bounds for the Distribution of Triangulated Surfaces in Moduli Space

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Abstract

Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the genus tends to infinity. Mirzakhani raised this question in her 2010 ICM address. We show that in the large genus case, triangulated surfaces are well distributed in moduli space in a fairly strong sense. We do this by proving upper and lower bounds for the number of triangulated surfaces lying in a Teichmüller ball in moduli space. In particular, we show that the number of triangulated surfaces lying in a Teichmüller unit ball is at most exponential in the number of triangles, independent of the genus.

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References

  1. Ahlfors, L.V.: Lectures on Quasiconformal Mappings. Van Nostrand, Princeton (1966)

    Google Scholar 

  2. Bender, E.A., Canfield, E.R.: The asymptotic number of rooted maps on a surface. J. Comb. Theory, Ser. A 43(2), 244–257 (1986)

    MathSciNet  Google Scholar 

  3. Bishop, C.J.: True trees are dense. Invent. Math. 197, 433–452 (2014)

    ADS  MathSciNet  Google Scholar 

  4. Bishop, C.J.: Constructing entire functions by quasiconformal folding. Acta Math. 214(1), 1–60 (2015)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  6. Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston (2010)

    Google Scholar 

  7. Buser, P., Sarnak, P.: On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117(1), 27–56 (1994)

    ADS  MathSciNet  Google Scholar 

  8. Budzinski, T., Curien, N., Petri, B.: Universality for random surfaces in unconstrained genus. Electron. J. Comb. 26(4), Paper 4.2 (2019)

    MathSciNet  Google Scholar 

  9. Budzinski, T., Curien, N., Petri, B.: The diameter of random Belyi surfaces. Algebraic Geom. Topol. 21(6), 2929–2957 (2021)

    MathSciNet  Google Scholar 

  10. Budzinski, T., Louf, B.: Local limits of uniform triangulations in high genus. Invent. Math. 223, 1–47 (2021)

    ADS  MathSciNet  Google Scholar 

  11. Budzinski, T., Curien, N., Petri, B.: On the minimal diameter of closed hyperbolic surfaces. Duke Math. J. 170(2), 365–377 (2021)

    MathSciNet  Google Scholar 

  12. Balacheff, F., Parlier, H., Sabourau, S.: Short loop decompositions of surfaces and the geometry of Jacobians. Geom. Funct. Anal. 22, 37–73 (2012)

    MathSciNet  Google Scholar 

  13. Cavendish, W., Parlier, H.: Growth of the Weil-Petersson diameter of moduli space. Duke Math. J. 161(1), 139–171 (2012)

    MathSciNet  Google Scholar 

  14. Curien, N.: Planar stochastic hyperbolic triangulations. Probab. Theory Relat. Fields 165(3), 509–540 (2016)

    MathSciNet  Google Scholar 

  15. Demailly, J.P.: Complex Analytic and Differential Geometry (2009). https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf

    Google Scholar 

  16. Douady, A., Earle, C.J.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986)

    MathSciNet  Google Scholar 

  17. Eskin, A., Okounkov, A.: Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145(1), 59–103 (2001)

    ADS  MathSciNet  Google Scholar 

  18. Fletcher, A., Kahn, J., Markovic, V.: The moduli space of Riemann surfaces of large genus. Geom. Funct. Anal. 23, 867–887 (2013)

    MathSciNet  Google Scholar 

  19. Gamburd, A.: Poisson-Dirichlet distribution for random Belyi surfaces. Ann. Probab. 34(5), 1827–1848 (2006)

    MathSciNet  Google Scholar 

  20. Gardiner, F.: Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York (1987)

    Google Scholar 

  21. Grushevsky, S.: An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces. Math. Ann. 321, 1–13 (2001)

    MathSciNet  Google Scholar 

  22. Guth, L., Parlier, H., Young, R.: Pants decompositions of random surfaces. Geom. Funct. Anal. 21, 1069 (2011)

    MathSciNet  Google Scholar 

  23. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

    Google Scholar 

  24. Hubbard, J.H.: Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Volume 1: Teichmüller Theory. Matrix Editions, Ithaca (2006)

    Google Scholar 

  25. Jenkins, J.A.: On the existence of certain extremal metrics. Ann. Math. 66, 440453 (1957)

    MathSciNet  Google Scholar 

  26. Kerckhoff, S.P.: The asymptotic geometry of Teichmüller space. Topology 19(1), 23–41 (1980)

    MathSciNet  Google Scholar 

  27. Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. World Scientific, Hackensack (2005)

    Google Scholar 

  28. Le Gall, J.-F.: The topological structure of scaling limits of large planar maps. Invent. Math. 169(3), 621–670 (2007)

    ADS  MathSciNet  Google Scholar 

  29. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, Berlin (1973)

    Google Scholar 

  30. Lipnowski, M., Wright, A.: Towards optimal spectral gaps in large genus (2021)

  31. Marzouk, C.: Scaling limits of random bipartite planar maps with a prescribed degree sequence. Random Struct. Algorithms 53(3), 448–503 (2018)

    MathSciNet  Google Scholar 

  32. McMullen, C.: The moduli space of Riemann surfaces is Kahler hyperbolic. Ann. Math. 151(1), 327–357 (2000)

    MathSciNet  Google Scholar 

  33. Miermont, G.: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210(2), 319–401 (2013)

    MathSciNet  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  40. Monk, L.: Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus (2020)

  41. Monk, L., Thomas, J.: The tangle-free hypothesis on random hyperbolic surfaces (2020)

  42. Nie, X., Wu, Y., Xue, Y.: Large genus asymptotics for lengths of separating closed geodesics on random surfaces (2020)

  43. Penner, R.C.: Weil-Petersson volumes. J. Differ. Geom. 35(3), 559–608 (1992)

    MathSciNet  Google Scholar 

  44. Royden, H.L.: Automorphisms and isometries of Teichmilller space. In: Advances in the Theory of Riemann Surfaces. Annals of Mathematics Studies, vol. 66. Princeton University Press, Princeton (1971)

    Google Scholar 

  45. Strebel, K.: Uber quadratische differentiate mit geschlossen trajectorien und extremale quasikonforme abbildungen. In: Festband zum 70 Geburstag von Rolf Nevanlinna. Springer, Berlin (1966)

    Google Scholar 

  46. Vaisala, J.: Lectures on n-Dimensional Quasiconformal Mappings. Springer, Berlin (1971)

    Google Scholar 

  47. Wright, A.: Translation surfaces and their orbit closures: an introduction for a broad audience. EMS Surv. Math. Sci. 2, 63–108 (2015)

    MathSciNet  Google Scholar 

  48. Wu, Y., Xue, Y.: Small eigenvalues of closed Riemann surfaces for large genus (2018)

  49. Wu, Y., Xue, Y.: Random hyperbolic surfaces of large genus have first eigenvalues greater than \(\frac{3}{16}-\varepsilon \). Geom. Funct. Anal. 32, 340–410 (2022)

    MathSciNet  Google Scholar 

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Acknowledgements

I thank my advisor Larry Guth for many inspiring discussions and enormous help with this paper. I thank Curt McMullen for many comments on the geometry of Teichmüller space. I thank Chris Bishop for correspondence on quasiconformal maps. I have learned a lot from conversations with Morris Ang, Scott Sheffield and Yilin Wang on random triangulations in probability theory. I thank the referee for communicating to me the proof of Proposition 3.9 due to Maxime Fortier Bourque. I thank Robert Burklund, Yilin Wang and the referee for helping me improve the writing.

Funding

This research was supported by the National Science Foundation Graduate Research Fellowship Program (under Grant No. 1745302) and the Simons Foundation (under Larry Guth’s Simons Investigator award).

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Sahana Vasudevan

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  1. Sahana Vasudevan
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Correspondence to Sahana Vasudevan.

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Vasudevan, S. Large Genus Bounds for the Distribution of Triangulated Surfaces in Moduli Space. Geom. Funct. Anal. 34, 529–630 (2024). https://doi.org/10.1007/s00039-023-00656-5

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