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Counting saddle connections in a homology class modulo \( q \) (with an appendix by Rodolfo Gutiérrez-Romo). (English) Zbl 1434.32020

Summary: We give effective estimates for the number of saddle connections on a translation surface that have length \( \leq L \) and are in a prescribed homology class modulo \( q \). Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
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[1] J. S. Athreya; Y. Cheung; H. Masur, Siegel-Veech transforms are in \(L^2\), J. Mod. Dyn., 14, 1-19 (2019) · Zbl 1432.32011
[2] A. Avila; S. Gouëzel; J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104, 143-211 (2006) · Zbl 1263.37051 · doi:10.1007/s10240-006-0001-5
[3] A. Avila; C. Matheus; J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups, Math. Ann., 370, 785-809 (2018) · Zbl 1381.05088 · doi:10.1007/s00208-017-1568-5
[4] A. Avila; M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198, 1-56 (2007) · Zbl 1143.37001 · doi:10.1007/s11511-007-0012-1
[5] B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. · Zbl 1146.22009
[6] E. Breuillard and H. Oh, editors, Thin Groups and Superstrong Approximation, Mathematical Sciences Research Institute Publications, 61, Cambridge University Press, Cambridge, 2014; Selected expanded papers from the workshop held in Berkeley, CA, February 6-10, 2012. · Zbl 1326.20001
[7] M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413, 36-67 (1991) · Zbl 0704.22009 · doi:10.1515/crll.1991.413.36
[8] A. Eskin; G. Margulis; S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147, 93-141 (1998) · Zbl 0906.11035 · doi:10.2307/120984
[9] A. Eskin; H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21, 443-478 (2001) · Zbl 1096.37501 · doi:10.1017/S0143385701001225
[10] A. Eskin; H. Masur; A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97, 61-179 (2003) · Zbl 1037.32013 · doi:10.1007/s10240-003-0015-1
[11] A. Eskin; M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5, 71-105 (2011) · Zbl 1219.37006 · doi:10.3934/jmd.2011.5.71
[12] R. Gutiérrez-Romo, Classification of Rauzy-Veech groups: Proof of the Zorich conjecture, Invent. Math., 215, 741-778 (2019) · Zbl 1420.37020 · doi:10.1007/s00222-018-0836-7
[13] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., 1, 71-74 (1967) · Zbl 0168.27602
[14] B. Kirkwood; B. R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math., 92, 111-125 (1981) · Zbl 0466.20023 · doi:10.2140/pjm.1981.92.111
[15] M. Kontsevich; A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153, 631-678 (2003) · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x
[16] A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. · Zbl 1183.22001
[17] M. Magee, On Selberg’s Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018. · Zbl 1439.11132
[18] G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, PWN, Warsaw, 1989,399-409. · Zbl 0689.10026
[19] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115, 169-200 (1982) · Zbl 0497.28012 · doi:10.2307/1971341
[20] H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988,215-228. · Zbl 0661.30034
[21] H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10, 151-176 (1990) · Zbl 0706.30035 · doi:10.1017/S0143385700005459
[22] A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017. · Zbl 1451.37049
[23] A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61, 269-298 (2014) · Zbl 1341.20042
[24] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115, 201-242 (1982) · Zbl 0486.28014 · doi:10.2307/1971391
[25] W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124, 441-530 (1986) · Zbl 0658.32016 · doi:10.2307/2007091
[26] W. A. Veech, Siegel measures, Ann. of Math. (2), 148, 895-944 (1998) · Zbl 0922.22003 · doi:10.2307/121033
[27] J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1-69. · Zbl 1248.37038
[28] A. Zorich, How do the leaves of a closed 1-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2,197, Amer. Math. Soc., Providence, RI, 1999,135-178. · Zbl 0976.37012
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