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

Erwan Lanneau

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 1, page 1-56
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Lanneau, Erwan. "Connected components of the strata of the moduli spaces of quadratic differentials." Annales scientifiques de l'École Normale Supérieure 41.1 (2008): 1-56. <http://eudml.org/doc/272137>.

@article{Lanneau2008,
abstract = {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.},
author = {Lanneau, Erwan},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {quadratic differentials; moduli space; measured foliations; teichmüller geodesic flow; measured foliation; Teichmüller geodesic flow},
language = {eng},
number = {1},
pages = {1-56},
publisher = {Société mathématique de France},
title = {Connected components of the strata of the moduli spaces of quadratic differentials},
url = {http://eudml.org/doc/272137},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Lanneau, Erwan
TI - Connected components of the strata of the moduli spaces of quadratic differentials
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 1
SP - 1
EP - 56
AB - 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.
LA - eng
KW - quadratic differentials; moduli space; measured foliations; teichmüller geodesic flow; measured foliation; Teichmüller geodesic flow
UR - http://eudml.org/doc/272137
ER -

References

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