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

Annales scientifiques de l'École Normale Supérieure (2008)

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

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topLanneau, 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

top- [1] C. Boissy & E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, preprint arXiv:0710.5614, 2007. Zbl1195.37030
- [2] A. Douady & J. Hubbard, On the density of Strebel differentials, Invent. Math.30 (1975), 175–179. Zbl0371.30017
- [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. Zbl1037.32013
- [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. Zbl0415.30038
- [6] M. Keane, Interval exchange transformations, Math. Z.141 (1975), 25–31. Zbl0278.28010MR357739
- [7] S. Kerckhoff, H. Masur & J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math.124 (1986), 293–311. Zbl0637.58010
- [8] M. Kontsevich & A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math.153 (2003), 631–678. Zbl1087.32010
- [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. Zbl1161.30033MR2423309
- [11] E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv.79 (2004), 471–501. Zbl1054.32007MR2081723
- [12] E. Lanneau, Parity of the spin structure defined by a quadratic differential, Geom. Topol.8 (2004), 511–538. Zbl1064.32010MR2057772
- [13] H. Masur, The Jenkins-Strebel differentials with one cylinder are dense, Comment. Math. Helv.54 (1979), 179–184. Zbl0407.30036MR535053
- [14] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169–200. Zbl0497.28012MR644018
- [15] H. Masur & J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv.68 (1993), 289–307. Zbl0792.30030
- [16] H. Masur & S. Tabachnikov, Rational billiards and flat structures, in Handbook of dynamical systems, Vol. 1A, North-Holland, 2002, 1015–1089. Zbl1057.37034
- [17] H. Masur & A. Zorich, Moduli spaces of quadratic differentials: The principal boundary, counting problems and the Siegel–Veech constants, GAFA to appear. Zbl1037.32013
- [18] G. Rauzy, Échanges d’intervalles et transformations induites, Acta Arith.34 (1979), 315–328. Zbl0414.28018MR543205
- [19] K. Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 5, Springer, 1984. Zbl0547.30001MR743423
- [20] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431. Zbl0674.57008MR956596
- [21] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math.115 (1982), 201–242. Zbl0486.28014MR644019
- [22] W. A. Veech, The Teichmüller geodesic flow, Ann. of Math.124 (1986), 441–530. Zbl0658.32016MR866707
- [23] W. A. Veech, Moduli spaces of quadratic differentials, J. Analyse Math.55 (1990), 117–171. Zbl0722.30032MR1094714
- [24] A. Zorich, Flat surfaces, in Frontiers in number theory, physics, and geometry. I, Springer, 2006, 437–583. Zbl1129.32012MR2261104
- [25] A. Zorich, Explicit Jenkins–Strebel representatives of all strata of Abelian and quadratic differentials, Journal of Modern Dynamics, to appear. Zbl1149.30033MR2366233
- [26] A. Zorich, Rauzy-Veech induction, Rauzy classes, generalized permutations on Mathematica, http://perso.univ-rennes1.fr/anton.zorich.

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