Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol q $\end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

Michael Magee; Rene Rühr

doi:10.3934/jmd.2019020

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2019, 15: 237-262. doi: 10.3934/jmd.2019020

Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

Michael Magee ^1, and Rene Rühr ^2,

1.	Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham DH1 3LE, United Kingdom
2.	Faculty of Mathematics, Technion, Haifa, 32000 Israel

(with an appendix by Rodolfo Gutiérrez-Romo)
Rodolfo Gutiérrez-Romo < rodolfo.gutierrez@imj-prg.fr>: Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586, Bátiment Sophie Germain, 75205 PARIS Cedex 13, France

Received November 30, 2018 Revised May 10, 2019 Published August 2019

Fund Project: Supported in part by the S.N.F., project number 168823

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.

Keywords: Saddle connections, effective counting, congruences, strong approximation, spectral gap.

Mathematics Subject Classification: Primary: 32G15, 11N45; Secondary: 37E35, 20H05, 37A30.

Citation: Michael Magee, Rene Rühr. Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo). Journal of Modern Dynamics, 2019, 15: 237-262. doi: 10.3934/jmd.2019020

References:

[1]	J. S. Athreya, Y. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19. Google Scholar
[2]	A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. Google Scholar
[3]	A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups, Math. Ann., 370 (2018), 785-809. doi: 10.1007/s00208-017-1568-5. Google Scholar
[4]	A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. Google Scholar
[5]	B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749. Google Scholar
[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. Google Scholar
[7]	M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67. doi: 10.1515/crll.1991.413.36. Google Scholar
[8]	A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar
[9]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. Google Scholar
[10]	A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1. Google Scholar
[11]	A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71. Google Scholar
[12]	R. Gutiérrez-Romo, Classification of Rauzy–Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741-778. doi: 10.1007/s00222-018-0836-7. Google Scholar
[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 (1967), 71-74. Google Scholar
[14]	B. Kirkwood and B. R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math., 92 (1981), 111-125. doi: 10.2140/pjm.1981.92.111. Google Scholar
[15]	M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x. Google Scholar
[16]	A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4. Google Scholar
[17]	M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018. Google Scholar
[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. Google Scholar
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar
[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. doi: 10.1007/978-1-4613-9602-4_20. Google Scholar
[21]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459. Google Scholar
[22]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017. Google Scholar
[23]	A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298. Google Scholar
[24]	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
[25]	W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091. Google Scholar
[26]	W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033. Google Scholar
[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. Google Scholar
[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. doi: 10.1090/trans2/197/05. Google Scholar

show all references

References:

[1]	J. S. Athreya, Y. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19. Google Scholar
[2]	A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. Google Scholar
[3]	A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups, Math. Ann., 370 (2018), 785-809. doi: 10.1007/s00208-017-1568-5. Google Scholar
[4]	A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. Google Scholar
[5]	B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749. Google Scholar
[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. Google Scholar
[7]	M. Burger, Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67. doi: 10.1515/crll.1991.413.36. Google Scholar
[8]	A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar
[9]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. Google Scholar
[10]	A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179. doi: 10.1007/s10240-003-0015-1. Google Scholar
[11]	A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71-105. doi: 10.3934/jmd.2011.5.71. Google Scholar
[12]	R. Gutiérrez-Romo, Classification of Rauzy–Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741-778. doi: 10.1007/s00222-018-0836-7. Google Scholar
[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 (1967), 71-74. Google Scholar
[14]	B. Kirkwood and B. R. McDonald, The symplectic group over a ring with one in its stable range, Pacific J. Math., 92 (1981), 111-125. doi: 10.2140/pjm.1981.92.111. Google Scholar
[15]	M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x. Google Scholar
[16]	A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010. doi: 10.1007/978-3-0346-0332-4. Google Scholar
[17]	M. Magee, On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, arXiv: 1609.05500, 2018. Google Scholar
[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. Google Scholar
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar
[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. doi: 10.1007/978-1-4613-9602-4_20. Google Scholar
[21]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459. Google Scholar
[22]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, arXiv: 1708.06263, 2017. Google Scholar
[23]	A. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation, 61 (2014), 269-298. Google Scholar
[24]	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
[25]	W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530. doi: 10.2307/2007091. Google Scholar
[26]	W. A. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033. Google Scholar
[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. Google Scholar
[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. doi: 10.1090/trans2/197/05. Google Scholar

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Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

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Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

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