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. AthreyaA. BufetovA. 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.

[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.

[3]

F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.

[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.

[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.

[6]

V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Enumeration of meanders and Masur–Veech volumes, arXiv e-prints, 2017, arXiv: 1705.05190.

[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.

[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.

[9]

V. Erlandsson, H. Parlier and J. Souto, Counting curves, and the stable length of currents, arXiv e-prints, 2016, arXiv: 1612.05980.

[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.

[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.

[12]

V. Erlandsson and C. Uyanik, Length functions on currents and applications to dynamics and counting, arXiv e-prints, 2018, arXiv: 1803.10801.

[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.

[14]

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012.

[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.

[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.

[17]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.

[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.

[19]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.

[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.

[21]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135.  doi: 10.1016/0040-9383(83)90023-X.

[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.

[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.

[24]

B. Martelli, An introduction to geometric topology, arXiv e-prints, 2016, arXiv: 1610.02592.

[25]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169–200. doi: 10.2307/1971341.

[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.

[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.

[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.

[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.

[30]

M. Mirzakhani, Counting Mapping Class group orbits on hyperbolic surfaces, arXiv e-prints, 2016, arXiv: 1601.03342.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

U. Wolf, The action of the mapping class group on the pants complex, preprint, 2009.

[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.

[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.

[40]

U. Wolf, Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex, Ph.D. thesis, 2009.

show all references

References:
[1]

J. AthreyaA. BufetovA. 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.

[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.

[3]

F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162.  doi: 10.1007/BF01393996.

[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.

[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.

[6]

V. Delecroix, E. Goujard, P. Zograf and A. Zorich, Enumeration of meanders and Masur–Veech volumes, arXiv e-prints, 2017, arXiv: 1705.05190.

[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.

[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.

[9]

V. Erlandsson, H. Parlier and J. Souto, Counting curves, and the stable length of currents, arXiv e-prints, 2016, arXiv: 1612.05980.

[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.

[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.

[12]

V. Erlandsson and C. Uyanik, Length functions on currents and applications to dynamics and counting, arXiv e-prints, 2018, arXiv: 1803.10801.

[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.

[14]

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012.

[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.

[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.

[17]

J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math., 142 (1979), 221-274.  doi: 10.1007/BF02395062.

[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.

[19]

S. P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology, 19 (1980), 23-41.  doi: 10.1016/0040-9383(80)90029-4.

[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.

[21]

G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), 119-135.  doi: 10.1016/0040-9383(83)90023-X.

[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.

[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.

[24]

B. Martelli, An introduction to geometric topology, arXiv e-prints, 2016, arXiv: 1610.02592.

[25]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169–200. doi: 10.2307/1971341.

[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.

[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.

[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.

[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.

[30]

M. Mirzakhani, Counting Mapping Class group orbits on hyperbolic surfaces, arXiv e-prints, 2016, arXiv: 1601.03342.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

U. Wolf, The action of the mapping class group on the pants complex, preprint, 2009.

[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.

[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.

[40]

U. Wolf, Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex, Ph.D. thesis, 2009.

Figure 1.  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} $
Figure 2.  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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