Volumes of strata of moduli spaces of quadratic differentials: getting explicit values
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2203-2251.

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article applies all known approaches to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all dimensions up to 10.

Les volumes de strates de différentielles abéliennes ou quadratiques jouent un rôle important dans l’étude de la dynamique sur les surfaces plates, en lien avec la dynamique des billards polygonaux. Dans cet article nous utilisons toutes les approches connues pour calculer les volumes dans le cas quadratique et fournissons des valeurs explicites pour les volumes de toutes les strates de différentielles quadratiques méromorphes à pôles au plus simples jusqu’en dimension 10.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3062
Classification: 30F30, 14N10, 32G15
Keywords: flat surfaces, quadratic differentials, volumes, strata
Mot clés : surfaces plates, différentielles quadratiques, volumes, strates
Goujard, Elise 1

1 Laboratoire de Mathématiques d’Orsay 137 Rue de Paris 91400 Orsay (France)
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Goujard, Elise. Volumes of strata of moduli spaces of quadratic differentials: getting explicit values. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2203-2251. doi : 10.5802/aif.3062. http://www.numdam.org/articles/10.5802/aif.3062/

[1] Athreya, Jayadev S.; Eskin, Alex; Zorich, Anton Right-Angled Billiards and Volumes of Moduli Spaces of Quadratic Differentials on P 1 (http://arxiv.org/abs/1212.1660)

[2] Athreya, Jayadev S.; Eskin, Alex; Zorich, Anton Counting generalized Jenkins-Strebel differentials, Geom. Dedicata, Volume 170 (2014), pp. 195-217 | DOI

[3] Boissy, Corentin Configurations of saddle connections of quadratic differentials on ℂℙ 1 and on hyperelliptic Riemann surfaces, Comment. Math. Helv., Volume 84 (2009) no. 4, pp. 757-791 | DOI

[4] Chen, Dawei; Möller, Martin Quadratic differentials in low genus: exceptional and non-varying strata, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 2, pp. 309-369

[5] Delecroix, Vincent; Goujard, Elise; Zograf, P.; Zorich, Anton Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes (in progress)

[6] Delecroix, Vincent; Hubert, Pascal; Lelièvre, Samuel Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 6, pp. 1085-1110

[7] Delecroix, Vincent; Zorich, Anton Cries and whispers in wind-tree forests (http://arxiv.org/abs/1502.06405)

[8] Eskin, Alex; Kontsevich, Maxim; Zorich, Anton Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci., Volume 120 (2014), pp. 207-333 | DOI

[9] Eskin, Alex; Masur, Howard; Zorich, Anton Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. (2003) no. 97, pp. 61-179 | DOI

[10] Eskin, Alex; Okounkov, Andrei Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., Volume 145 (2001) no. 1, pp. 59-103 | DOI

[11] Eskin, Alex; Okounkov, Andrei Pillowcases and quasimodular forms, Algebraic geometry and number theory (Progr. Math.), Volume 253, Birkhäuser Boston, Boston, MA, 2006, pp. 1-25 | DOI

[12] Goujard, Elise Table of volumes of strata of quadratic differentials (up to dimension 11) (https://sites.google.com/site/elisegoujard/home/recherche-research/tablevoljune2015.pdf)

[13] Goujard, Elise Siegel-Veech constants for strata of moduli spaces of quadratic differentials, Geom. Funct. Anal., Volume 25 (2015) no. 5, pp. 1440-1492 | DOI

[14] Jackson, David M.; Visentin, Terry I. An atlas of the smaller maps in orientable and nonorientable surfaces, CRC Press Series on Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2001, viii+279 pages

[15] Kontsevich, Maxim Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 1-23 http://projecteuclid.org/euclid.cmp/1104250524 | DOI

[16] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | DOI

[17] Lanneau, Erwan Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv., Volume 79 (2004) no. 3, pp. 471-501 | DOI

[18] Lanneau, Erwan Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 1, pp. 1-56

[19] Lassalle, Michel An explicit formula for the characters of the symmetric group, Math. Ann., Volume 340 (2008) no. 2, pp. 383-405 | DOI

[20] Masur, Howard; Zorich, Anton Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., Volume 18 (2008) no. 3, pp. 919-987 | DOI

[21] Mirzakhani, Maryam Ergodic theory of the earthquake flow, Int. Math. Res. Not. IMRN (2008) no. 3, Art. ID rnm116, 39 pages | DOI

[22] Ríos-Zertuche, Rodolfo The pillowcase distribution and near-involutions, Electron. J. Probab., Volume 19 (2014), no. 116, 22 pages | DOI

[23] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages

[24] Zorich, Anton Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials, Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 459-471

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