The distribution of critical graphs of Jenkins–Strebel differentials

Arana-Herrera, Francisco¹ ; Calderon, Aaron²

¹Department of Mathematics, University of Maryland, College Park, MD, United States
²Department of Mathematics, University of Chicago, Chicago, IL, United States

Geometry & topology, Tome 29 (2025) no. 5, pp. 2571-2608

Résumé

By work of Jenkins and Strebel, given a Riemann surface X and a simple closed multicurve α on it, there exists a unique quadratic differential q on X whose horizontal foliation is measure equivalent to α. We study the distribution of the critical graphs of these differentials in the moduli space of metric ribbon graphs as the extremal length of the multicurves goes to infinity, showing they equidistribute to the Kontsevich measure regardless of the initial choice of X.

Détail

DOI : 10.2140/gt.2025.29.2571

Keywords: critical graph, quadratic differential, Jenkins, Strebel, equidistribution, metric ribbon graph

Bibliographie

[1] M Aka, M Einsiedler, U Shapira, Integer points on spheres and their orthogonal grids, J. Lond. Math. Soc. 93 (2016) 143 | DOI

[2] M Aka, M Einsiedler, U Shapira, Integer points on spheres and their orthogonal lattices, Invent. Math. 206 (2016) 379 | DOI

[3] F Arana-Herrera, Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani’s asymptotics for simple closed hyperbolic geodesics, J. Mod. Dyn. 16 (2020) 81 | DOI

[4] F Arana-Herrera, Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces, Geom. Dedicata 210 (2021) 65 | DOI

[5] F Arana-Herrera, Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes, Geom. Topol. 26 (2022) 1291 | DOI

[6] F Arana-Herrera, Effective mapping class group dynamics, I : Counting lattice points in Teichmüller space, Duke Math. J. 172 (2023) 1437 | DOI

[7] F Arana-Herrera, Effective mapping class group dynamics, III : Counting filling closed curves on surfaces, Camb. J. Math. 12 (2024) 563 | DOI

[8] F Arana-Herrera, Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata, Ergodic Theory Dynam. Systems 45 (2025) 649 | DOI

[9] F Arana-Herrera, A Calderon, The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics, preprint (2022)

[10] J Athreya, A Bufetov, A Eskin, M Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J. 161 (2012) 1055 | DOI

[11] A Avila, S Gouëzel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math. 178 (2013) 385 | DOI

[12] A Avila, S Gouëzel, J C Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006) 143 | DOI

[13] A Calderon, J Farre, Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow, (2024)

[14] A Calderon, J Farre, Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow, Geom. Topol. 28 (2024) 1995 | DOI

[15] N Do, The asymptotic Weil–Petersson form and intersection theory on Mg,n, preprint (2010)

[16] B Dozier, J Sapir, Coarse density of subsets of moduli space, Ann. Inst. Fourier (Grenoble) 71 (2021) 1121 | DOI

[17] D Dumas, Skinning maps are finite-to-one, Acta Math. 215 (2015) 55 | DOI

[18] M Einsiedler, R Rühr, P Wirth, Distribution of shapes of orthogonal lattices, Ergodic Theory Dynam. Systems 39 (2019) 1531 | DOI

[19] V Erlandsson, J Souto, Mirzakhani’s curve counting and geodesic currents, 345, Birkhäuser (2022) | DOI

[20] A Eskin, M Mirzakhani, Invariant and stationary measures for the SL(2, R) action on moduli space, Publ. Math. Inst. Hautes Études Sci. 127 (2018) 95 | DOI

[21] A Eskin, M Mirzakhani, A Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2, R) action on moduli space, Ann. of Math. 182 (2015) 673 | DOI

[22] A Eskin, M Mirzakhani, A Mohammadi, Effective counting of simple closed geodesics on hyperbolic surfaces, J. Eur. Math. Soc. 24 (2022) 3059 | DOI

[23] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[24] G Forni, Limits of geodesic push-forwards of horocycle invariant measures, Ergodic Theory Dynam. Systems 41 (2021) 2782 | DOI

[25] F P Gardiner, H Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991) 209 | DOI

[26] J Harer, D Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 457 | DOI

[27] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221 | DOI

[28] J A Jenkins, On the existence of certain general extremal metrics, Ann. of Math. 66 (1957) 440 | DOI

[29] S P Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980) 23 | DOI

[30] M Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1 | DOI

[31] M Liu, Length statistics of random multicurves on closed hyperbolic surfaces, Groups Geom. Dyn. 16 (2022) 437 | DOI

[32] F Luo, On Teichmüller spaces of surfaces with boundary, Duke Math. J. 139 (2007) 463 | DOI

[33] G A Margulis, On some aspects of the theory of Anosov systems, PhD thesis, Moscow State University (1970)

[34] H Masur, J Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. 134 (1991) 455 | DOI

[35] M Mirzakhani, Weil–Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007) 1 | DOI

[36] M Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. 2008 (2008) | DOI

[37] M Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. 168 (2008) 97 | DOI

[38] M Mirzakhani, Counting mapping class group orbits on hyperbolic surfaces, preprint (2016)

[39] G Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, from: "Handbook of Teichmüller theory, II", IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc. (2009) 151 | DOI

[40] G Mondello, Triangulated Riemann surfaces with boundary and the Weil–Petersson Poisson structure, J. Differential Geom. 81 (2009) 391

[41] P Norbury, Counting lattice points in the moduli space of curves, Math. Res. Lett. 17 (2010) 467 | DOI

[42] R C Penner, J L Harer, Combinatorics of train tracks, 125, Princeton Univ. Press (1992) | DOI

[43] H L Royden, Automorphisms and isometries of Teichmüller space, from: "Advances in the theory of Riemann surfaces", Ann. of Math. Stud. 66, Princeton Univ. Press (1971) 369 | DOI

[44] K Strebel, Über quadratische Differentiale mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen, from: "Festband zum 70 Geburtstag von Rolf Nevanlinna", Springer (1966) 105 | DOI

[45] K Strebel, On quadratic differentials and extremal quasi-conformal mappings, from: "Proceedings of the International Congress of Mathematicians, II" (editor R D James), Canad. Math. Congr. (1975) 223

[46] K Strebel, On the existence of extremal Teichmueller mappings, J. Anal. Math. 30 (1976) 464 | DOI