Connected components of the strata of the moduli spaces of quadratic differentials

Lanneau, Erwan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 1, pp. 1-56.

Dans des travaux maintenant classiques, Masur [14] et Veech [21] ont démontré indépendamment que le flot géodésique de Teichmüller est ergodique sur chaque composante connexe de chaque strate de l'espace des modules des différentielles quadratiques. Il devient dès lors intéressant d'avoir une description de ces composantes ergodiques. Veech a montré que ces strates ne sont pas nécessairement connexes. Dans un article récent, Kontsevich et Zorich [8] donnent une description complète des composantes dans le cas particulier où les différentielles quadratiques sont données par le carré de différentielles abéliennes. Dans cet article, nous considérons le cas complémentaire. Dans un précédent article [11], nous montrions que les strates ne sont pas forcément connexes. Nous donnions une série de strates non-connexes possédant des composantes connexes hyperelliptiques. Dans cet article, nous démontrons le théorème général annoncé dans [11] : excepté quatre cas particuliers en petits genres, les strates de l'espace des modules des différentielles quadratiques ont au plus deux composantes connexes, les cas de non-connexité étant décrits exactement par [11] : une composante est hyperelliptique, l'autre non. Notre preuve repose principalement sur une nouvelle approche des différentielles quadratiques de type Jenkins-Strebel, à savoir la notion de permutations généralisées.

In two fundamental classical papers, Masur [14] and Veech [21] have independently proved that the Teichmüller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work [8], Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the complementary case. In a previous paper [11], we have described some particular components, namely the hyperelliptic connected components and showed that some strata are non-connected. In this paper, we give the general classification theorem: up to four exceptional cases in low genera, the strata of meromorphic quadratic differentials are either connected, or have exactly two connected components where one component is hyperelliptic, the other not. This result was announced in the paper [11]. Our proof is based on a new approach of the so-called Jenkins-Strebel differential. We will present and use the notion of generalized permutations.

[1] C. Boissy & E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, preprint arXiv:0710.5614, 2007. | Zbl

[2] A. Douady & J. Hubbard, On the density of Strebel differentials, Invent. Math. 30 (1975), 175-179. | Zbl

[3] A. Eskin, H. Masur & 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. | Numdam | Zbl

[4] A. Fathi, H. Masur & V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Soc. Math. France, 1979.

[5] J. Hubbard & H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221-274. | Zbl

[6] M. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25-31. | MR | Zbl

[7] S. Kerckhoff, H. Masur & J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. 124 (1986), 293-311. | Zbl

[8] M. Kontsevich & A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631-678. | Zbl

[9] M. Kontsevich & A. Zorich, Lyapunov exponents and Hodge theory, preprint arXiv:hep-th/9701164v1, 2007.

[10] E. Lanneau, Classification of connected components of the strata of the moduli spaces of quadratic differentials with prescribed singularities, Thèse, Université de Rennes 1, december 2003.

[11] E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv. 79 (2004), 471-501. | MR | Zbl

[12] E. Lanneau, Parity of the spin structure defined by a quadratic differential, Geom. Topol. 8 (2004), 511-538. | MR | Zbl

[13] H. Masur, The Jenkins-Strebel differentials with one cylinder are dense, Comment. Math. Helv. 54 (1979), 179-184. | MR | Zbl

[14] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982), 169-200. | MR | Zbl

[15] H. Masur & J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv. 68 (1993), 289-307. | Zbl

[16] H. Masur & S. Tabachnikov, Rational billiards and flat structures, in Handbook of dynamical systems, Vol. 1A, North-Holland, 2002, 1015-1089. | Zbl

[17] H. Masur & A. Zorich, Moduli spaces of quadratic differentials: The principal boundary, counting problems and the Siegel-Veech constants, GAFA to appear.

[18] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith. 34 (1979), 315-328. | MR | Zbl

[19] K. Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 5, Springer, 1984. | MR | Zbl

[20] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417-431. | MR | Zbl

[21] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. 115 (1982), 201-242. | MR | Zbl

[22] W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. 124 (1986), 441-530. | MR | Zbl

[23] W. A. Veech, Moduli spaces of quadratic differentials, J. Analyse Math. 55 (1990), 117-171. | MR | Zbl

[24] A. Zorich, Flat surfaces, in Frontiers in number theory, physics, and geometry. I, Springer, 2006, 437-583. | MR | Zbl

[25] A. Zorich, Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials, Journal of Modern Dynamics, to appear. | MR | Zbl

[26] A. Zorich, Rauzy-Veech induction, Rauzy classes, generalized permutations on Mathematica, http://perso.univ-rennes1.fr/anton.zorich.