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.

Classification : 32G15, 30F30, 57R30, 37D40
Mots clés : quadratic differentials, moduli space, measured foliations, teichmüller geodesic flow

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