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The local-global principle for integral Soddy sphere packings
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 |
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.
References:
[1] |
J. S. Athreya, Y. Cheung and H. Masur,
Siegel–Veech transforms are in $L^{2}$, J. Mod. Dyn., 14 (2019), 1-19.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
M. Burger,
Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67.
doi: 10.1515/crll.1991.413.36. |
[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. |
[9] |
A. Eskin and H. Masur,
Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.
doi: 10.1017/S0143385701001225. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[16] |
A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010.
doi: 10.1007/978-3-0346-0332-4. |
[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. |
[19] |
H. Masur,
Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.
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.
doi: 10.1007/978-1-4613-9602-4_20. |
[21] |
H. Masur,
The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[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.
|
[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. |
[25] |
W. A. Veech,
The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
doi: 10.2307/2007091. |
[26] |
W. A. Veech,
Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.
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. |
[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. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
M. Burger,
Kazhdan constants for SL(3, Z), J. Reine Angew. Math., 413 (1991), 36-67.
doi: 10.1515/crll.1991.413.36. |
[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. |
[9] |
A. Eskin and H. Masur,
Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.
doi: 10.1017/S0143385701001225. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[16] |
A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Springer Science and Business Media, 2010.
doi: 10.1007/978-3-0346-0332-4. |
[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. |
[19] |
H. Masur,
Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.
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.
doi: 10.1007/978-1-4613-9602-4_20. |
[21] |
H. Masur,
The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[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.
|
[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. |
[25] |
W. A. Veech,
The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
doi: 10.2307/2007091. |
[26] |
W. A. Veech,
Siegel measures, Ann. of Math. (2), 148 (1998), 895-944.
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. |
[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. |
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