Abstract
We present an explicit formula relating volumes of strata of meromorphic quadratic differentials with at most simple poles on Riemann surfaces and counting functions of the number of flat cylinders filled by closed geodesics in associated flat metric with singularities. This generalizes the result of Athreya, Eskin and Zorich in genus 0 to higher genera.
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Athreya J.S., Eskin A., Zorich A.: Counting generalized Jenkins–Strebel differentials. Geometriae Dedicata 170, 195–217 (2014)
J.S. Athreya, A. Eskin and A. Zorich. Right-Angled Billiards and Volumes of Moduli Spaces of Quadratic Differentials on \({\mathbb{C} P^1}\) (2015). arXiv:1212.1660
Bainbridge M.: Euler characteristics of Teichmüller curves in genus two. Geometry and Topology 11, 1887–2073 (2007)
Bainbridge M.: Billiards in L-shaped tables with barriers. GAFA 20(2), 299–356 (2010)
Bainbridge M.: Erratum to: Billiards in L-shaped tables with barriers. GAFA 20(5), 1306 (2010)
Boissy C.: Configurations of saddle connections of quadratic differentials on \({\mathbb{CP}^1}\) and on hyperelliptic Riemann surfaces. Commentarii Mathematici Helvetici, 84(4), 757–791 (2009)
Bauer M., Goujard E.: Geometry of periodic regions on flat surfaces and associated Siegel–Veech constants. Geometriae Dedicata 174(1), 203–233 (2015)
J. Chaika and A. Eskin. Every flat surface is Birkhoff and Osceledets generic in almost every direction, JMD (2014)
Chen D., Möller M.: Quadratic differentials in low genus: exceptional and non-varying. Annales scientifiques de l’ENS 47(2), 309–369 (2014)
V. Delecroix, E. Goujard, P. Zograf and A. Zorich. Square-tiled surfaces of fiwed combinatorial type: equidistribution, counting, volume of the ambient strata (2015, in preparation)
Delecroix V., Hubert P., Lelievre S.: Diffusion for the periodic wind-tree model. Annales de l’ENS 47(6), 1085–1110 (2014)
A. Eskin, M. Kontsevich and A. Zorich. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. In: Publications mathématiques de l’IHÉS, Vol. 120, No. 1. Springer, Berlin (2014), pp. 207–333
Eskin A., Masur H.: Asymptotic formulas on flat surfaces. Ergodic Theory and Dynamical Systems 21(2), 443–478 (2001)
Eskin A., Masur H., Zorich A.: Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publications de l’IHES 97(1), 61–179 (2003)
Eskin A., Okounkov A.: Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Inventiones Mathematicae 145(1), 59–104 (2001)
Eskin A., Okounkov A.: Pillowcases and quasimodular forms. Algebraic Geometry and Number Theory, Progress in Mathematics 253, 1–25 (2006)
Eskin A., Okounkov A., Pandharipande R.: The theta characteristic of a branched covering. Advances in Mathematics 217(3), 873–888 (2008)
S. Filip. Splitting mixed Hodge structures over affine invariant manifolds. arXiv:1311.2350 (2013)
Forni G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Annals of Mathematics 155(1), 1–103 (2002)
Forni G.: On the Lyapunov exponents of the Kontsevich–Zorich cocycle. In: Hasselblatt, B., Katok, A. (eds) Handbook of Dynamical Systems, Vol. 1B, pp. 549–580. Elsevier, Amsterdam (2006)
E. Goujard. Volumes of strata of moduli spaces of quadratic differentials: getting explicit values. arXiv:1501.01611
M. Kontsevich, Lyapunov exponents and Hodge theory. The mathematical beauty of physics (Saclay, 1996), (in Honor of C. Itzykson). In: Adv. Ser. Math. Phys., Vol. 24. World Sci. Publishing, River Edge, pp. 318–332 (1997)
Kontsevich M., Zorich A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Inventiones Mathematicae 153(3), 631–678 (2003)
Korotkin D., Zograf P.: Tau function and moduli of differentials. Mathematical Research Letters 18(3), 447–458 (2011)
Lanneau E.: Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities. Commentarii Mathematici Helvetici 79(3), 471–501 (2004)
Lanneau E.: Connected components of the strata of the moduli spaces of quadratic differentials. Annales Scientifiques de l’ENS 41(1), 1–56 (2008)
Masur H., Smillie J.: Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms. Commentarii Mathematici Helvetici 68, 289–307 (1993)
Masur H., Zorich A.: Multiple saddle connections on flat surfaces ans the principal boundary of the moduli spaces of quadratic differentials. Geometric and Functional Analysis 18(3), 919–987 (2008)
Mirzakhani M.: Growth of the number of simple closed geodesics on hyperbolic surfaces. Annals of Mathematics 168, 97–125 (2008)
Veech W.: Siegel measures. Annals of Mathematics 148, 895–944 (1998)
Vorobets Ya.: Periodic geodesics of translation surfaces. In: Kolyada, S., Manin, Yu. I., Ward, T. (eds) Algebraic and Topological Dynamics, Contemporary Mathematics, Vol. 385, pp. 205–258. Amer. Math. Soc., Providence (2005)
Zorich A.: Square tiled surfaces and Teichmüller volumes of the moduli spaces of Abelian differentials in collection. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry, pp. 459–471. Springer, Berlin (2002)
A. Zorich. Asymptotic flag of an orientable measured foliation on a surface. In: Geometric Study of Foliations. World Scientific Pb. Co. (1994), pp. 479–498
Zorich A.: How do the leaves of a closed 1-form wind around a surface. In: Arnold, V., Kontsevich, M., Zorich, A. (eds) Pseudoperiodic Topology, Translations of the AMS, Ser.2, Vol. 197, pp. 135–178. AMS, Providence (1999)
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Goujard, E. Siegel–Veech Constants for Strata of Moduli Spaces of Quadratic Differentials. Geom. Funct. Anal. 25, 1440–1492 (2015). https://doi.org/10.1007/s00039-015-0345-4
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DOI: https://doi.org/10.1007/s00039-015-0345-4