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)

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.

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. AthreyaY. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19.   Google Scholar

[2]

A. AvilaS. 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. AvilaC. 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. EskinG. 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. EskinH. 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. AthreyaY. Cheung and H. Masur, Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19.   Google Scholar

[2]

A. AvilaS. 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. AvilaC. 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. EskinG. 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. EskinH. 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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