# American Institute of Mathematical Sciences

2020, 16: 81-107. doi: 10.3934/jmd.2020004

## Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

 Department of Mathematics, Stanford University, 450 Jane Stanford Way, Stanford, CA 94305-2125, USA

Received  April 20, 2019 Revised  October 06, 2019 Published  April 2020

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

Citation: Francisco Arana-Herrera. Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics. Journal of Modern Dynamics, 2020, 16: 81-107. doi: 10.3934/jmd.2020004
##### References:
 [1] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111.  doi: 10.1215/00127094-1548443.  Google Scholar [2] J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\Bbb C\rm P^1$, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 1311–1386.  Google Scholar [3] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.  Google Scholar [4] F. Bonahon, Geodesic laminations on surfaces, in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, 1–37. doi: 10.1090/conm/269/04327.  Google Scholar [5] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Square-tiled surfaces of fixed combinatorial type: Equidistribution, counting, volumes of the ambient strata, arXiv e-prints, 2016, arXiv: 1612.08374. Google Scholar [6] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Enumeration of meanders and Masur–Veech volumes, arXiv e-prints, 2017, arXiv: 1705.05190. Google Scholar [7] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves, arXiv e-prints, 2019, arXiv: 1908.08611. Google Scholar [8] A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., 145 (2001), 59-103.  doi: 10.1007/s002220100142.  Google Scholar [9] V. Erlandsson, H. Parlier and J. Souto, Counting curves, and the stable length of currents, arXiv e-prints, 2016, arXiv: 1612.05980. Google Scholar [10] V. Erlandsson, A remark on the word length in surface groups, Trans. Amer. Math. Soc., 372 (2019), 441-455.  doi: 10.1090/tran/7561.  Google Scholar [11] V. Erlandsson and J. Souto, Counting curves in hyperbolic surfaces, Geom. Funct. Anal., 26 (2016), 729-777.  doi: 10.1007/s00039-016-0374-7.  Google Scholar [12] V. Erlandsson and C. Uyanik, Length functions on currents and applications to dynamics and counting, arXiv e-prints, 2018, arXiv: 1803.10801. Google Scholar [13] A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Translated from the 1979 French original by Djun M. Kim and Dan Margalit, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [14] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [15] F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.  Google Scholar [16] F. P. Gardiner and H. Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl., 16 (1991), 209-237.  doi: 10.1080/17476939108814480.  Google Scholar [17] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.  Google Scholar [18] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 2. Surface Homeomorphisms and Rational Functions, Matrix Editions, Ithaca, NY, 2016.  Google Scholar [19] S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar [20] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147 (1992), 1-23.  doi: 10.1007/BF02099526.  Google Scholar [21] G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135.  doi: 10.1016/0040-9383(83)90023-X.  Google Scholar [22] E. Lindenstrauss and M. Mirzakhani, Ergodic theory of the space of measured laminations, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnm126, 49pp. doi: 10.1093/imrn/rnm126.  Google Scholar [23] G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar [24] B. Martelli, An introduction to geometric topology, arXiv e-prints, 2016, arXiv: 1610.02592. Google Scholar [25] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169–200. doi: 10.2307/1971341.  Google Scholar [26] H. Masur, Ergodic actions of the mapping class group, Proc. Amer. Math. Soc., 94 (1985), 455-459.  doi: 10.1090/S0002-9939-1985-0787893-5.  Google Scholar [27] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222.  doi: 10.1007/s00222-006-0013-2.  Google Scholar [28] M. Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnm116, 39pp. doi: 10.1093/imrn/rnm116.  Google Scholar [29] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2), 168 (2008), 97–125. doi: 10.4007/annals.2008.168.97.  Google Scholar [30] M. Mirzakhani, Counting Mapping Class group orbits on hyperbolic surfaces, arXiv e-prints, 2016, arXiv: 1601.03342. Google Scholar [31] L. Monin and V. Telpukhovskiy, On normalizations of Thurston measure on the space of measured laminations, Topology Appl., 267 (2019), 106878, 12 pp. doi: 10.1016/j.topol.2019.106878.  Google Scholar [32] A. Papadopoulos, Geometric intersection functions and Hamiltonian flows on the space of measured foliations on a surface, Pacific J. Math., 124 (1986), 375-402.  doi: 10.2140/pjm.1986.124.375.  Google Scholar [33] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. doi: 10.1515/9781400882458.  Google Scholar [34] I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math., 231 (2012), 2391-2412.  doi: 10.1016/j.aim.2012.07.018.  Google Scholar [35] K. Rafi and J. Souto, Geodesic currents and counting problems, Geom. Funct. Anal., 29 (2019), 871-889.  doi: 10.1007/s00039-019-00502-7.  Google Scholar [36] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201–242. doi: 10.2307/1971391.  Google Scholar [37] U. Wolf, The action of the mapping class group on the pants complex, preprint, 2009. Google Scholar [38] S. Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math., 107 (1985), 969-997.  doi: 10.2307/2374363.  Google Scholar [39] M. Wolf, On realizing measured foliations via quadratic differentials of harmonic maps to $\mathbf R$-trees, J. Anal. Math., 68 (1996), 107-120.  doi: 10.1007/BF02790206.  Google Scholar [40] U. Wolf, Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex, Ph.D. thesis, 2009. Google Scholar

show all references

##### References:
 [1] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., 161 (2012), 1055-1111.  doi: 10.1215/00127094-1548443.  Google Scholar [2] J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\Bbb C\rm P^1$, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 1311–1386.  Google Scholar [3] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.  Google Scholar [4] F. Bonahon, Geodesic laminations on surfaces, in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, 1–37. doi: 10.1090/conm/269/04327.  Google Scholar [5] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Square-tiled surfaces of fixed combinatorial type: Equidistribution, counting, volumes of the ambient strata, arXiv e-prints, 2016, arXiv: 1612.08374. Google Scholar [6] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Enumeration of meanders and Masur–Veech volumes, arXiv e-prints, 2017, arXiv: 1705.05190. Google Scholar [7] V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves, arXiv e-prints, 2019, arXiv: 1908.08611. Google Scholar [8] A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., 145 (2001), 59-103.  doi: 10.1007/s002220100142.  Google Scholar [9] V. Erlandsson, H. Parlier and J. Souto, Counting curves, and the stable length of currents, arXiv e-prints, 2016, arXiv: 1612.05980. Google Scholar [10] V. Erlandsson, A remark on the word length in surface groups, Trans. Amer. Math. Soc., 372 (2019), 441-455.  doi: 10.1090/tran/7561.  Google Scholar [11] V. Erlandsson and J. Souto, Counting curves in hyperbolic surfaces, Geom. Funct. Anal., 26 (2016), 729-777.  doi: 10.1007/s00039-016-0374-7.  Google Scholar [12] V. Erlandsson and C. Uyanik, Length functions on currents and applications to dynamics and counting, arXiv e-prints, 2018, arXiv: 1803.10801. Google Scholar [13] A. Fathi, F. Laudenbach and V. Poénaru, Thurston's Work on Surfaces, Translated from the 1979 French original by Djun M. Kim and Dan Margalit, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [14] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar [15] F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.  Google Scholar [16] F. P. Gardiner and H. Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl., 16 (1991), 209-237.  doi: 10.1080/17476939108814480.  Google Scholar [17] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.  Google Scholar [18] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 2. Surface Homeomorphisms and Rational Functions, Matrix Editions, Ithaca, NY, 2016.  Google Scholar [19] S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.  Google Scholar [20] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147 (1992), 1-23.  doi: 10.1007/BF02099526.  Google Scholar [21] G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135.  doi: 10.1016/0040-9383(83)90023-X.  Google Scholar [22] E. Lindenstrauss and M. Mirzakhani, Ergodic theory of the space of measured laminations, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnm126, 49pp. doi: 10.1093/imrn/rnm126.  Google Scholar [23] G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar [24] B. Martelli, An introduction to geometric topology, arXiv e-prints, 2016, arXiv: 1610.02592. Google Scholar [25] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169–200. doi: 10.2307/1971341.  Google Scholar [26] H. Masur, Ergodic actions of the mapping class group, Proc. Amer. Math. Soc., 94 (1985), 455-459.  doi: 10.1090/S0002-9939-1985-0787893-5.  Google Scholar [27] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222.  doi: 10.1007/s00222-006-0013-2.  Google Scholar [28] M. Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnm116, 39pp. doi: 10.1093/imrn/rnm116.  Google Scholar [29] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2), 168 (2008), 97–125. doi: 10.4007/annals.2008.168.97.  Google Scholar [30] M. Mirzakhani, Counting Mapping Class group orbits on hyperbolic surfaces, arXiv e-prints, 2016, arXiv: 1601.03342. Google Scholar [31] L. Monin and V. Telpukhovskiy, On normalizations of Thurston measure on the space of measured laminations, Topology Appl., 267 (2019), 106878, 12 pp. doi: 10.1016/j.topol.2019.106878.  Google Scholar [32] A. Papadopoulos, Geometric intersection functions and Hamiltonian flows on the space of measured foliations on a surface, Pacific J. Math., 124 (1986), 375-402.  doi: 10.2140/pjm.1986.124.375.  Google Scholar [33] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. doi: 10.1515/9781400882458.  Google Scholar [34] I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math., 231 (2012), 2391-2412.  doi: 10.1016/j.aim.2012.07.018.  Google Scholar [35] K. Rafi and J. Souto, Geodesic currents and counting problems, Geom. Funct. Anal., 29 (2019), 871-889.  doi: 10.1007/s00039-019-00502-7.  Google Scholar [36] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201–242. doi: 10.2307/1971391.  Google Scholar [37] U. Wolf, The action of the mapping class group on the pants complex, preprint, 2009. Google Scholar [38] S. Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math., 107 (1985), 969-997.  doi: 10.2307/2374363.  Google Scholar [39] M. Wolf, On realizing measured foliations via quadratic differentials of harmonic maps to $\mathbf R$-trees, J. Anal. Math., 68 (1996), 107-120.  doi: 10.1007/BF02790206.  Google Scholar [40] U. Wolf, Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex, Ph.D. thesis, 2009. Google Scholar
Example of a quadratic differential in the principal stratum of $\textbf{Re}^{-1}([\gamma_1]) \subseteq Q\mathcal{M}_{2,0}$ for a (non-separating) simple closed curve $\gamma_1$ in $S_{2,0}$
No escape of mass property in the real period coordinate chart (b) associated to the polygon representation (a), representing a flat pillowcase in the principal stratum of $\mathrm{Re}^{-1}(\gamma_1) \subseteq Q\mathcal{T}_{0,4}$. The blue region covers $K_\epsilon$ and the gray region covers $\widehat{E}(\gamma_1) \backslash K_\epsilon$
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