American Institute of Mathematical Sciences

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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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