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

top
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

top

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

top
  1. [1] C. Boissy & E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, preprint arXiv:0710.5614, 2007. Zbl1195.37030
  2. [2] A. Douady & J. Hubbard, On the density of Strebel differentials, Invent. Math.30 (1975), 175–179. Zbl0371.30017
  3. [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. [4] A. Fathi, H. Masur & V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Soc. Math. France, 1979. 
  5. [5] J. Hubbard & H. Masur, Quadratic differentials and foliations, Acta Math.142 (1979), 221–274. Zbl0415.30038
  6. [6] M. Keane, Interval exchange transformations, Math. Z.141 (1975), 25–31. Zbl0278.28010MR357739
  7. [7] S. Kerckhoff, H. Masur & J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math.124 (1986), 293–311. Zbl0637.58010
  8. [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. [9] M. Kontsevich & A. Zorich, Lyapunov exponents and Hodge theory, preprint arXiv:hep-th/9701164v1, 2007. 
  10. [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. [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. [12] E. Lanneau, Parity of the spin structure defined by a quadratic differential, Geom. Topol.8 (2004), 511–538. Zbl1064.32010MR2057772
  13. [13] H. Masur, The Jenkins-Strebel differentials with one cylinder are dense, Comment. Math. Helv.54 (1979), 179–184. Zbl0407.30036MR535053
  14. [14] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169–200. Zbl0497.28012MR644018
  15. [15] H. Masur & J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv.68 (1993), 289–307. Zbl0792.30030
  16. [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. [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. [18] G. Rauzy, Échanges d’intervalles et transformations induites, Acta Arith.34 (1979), 315–328. Zbl0414.28018MR543205
  19. [19] K. Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 5, Springer, 1984. Zbl0547.30001MR743423
  20. [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. [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. [22] W. A. Veech, The Teichmüller geodesic flow, Ann. of Math.124 (1986), 441–530. Zbl0658.32016MR866707
  23. [23] W. A. Veech, Moduli spaces of quadratic differentials, J. Analyse Math.55 (1990), 117–171. Zbl0722.30032MR1094714
  24. [24] A. Zorich, Flat surfaces, in Frontiers in number theory, physics, and geometry. I, Springer, 2006, 437–583. Zbl1129.32012MR2261104
  25. [25] A. Zorich, Explicit Jenkins–Strebel representatives of all strata of Abelian and quadratic differentials, Journal of Modern Dynamics, to appear. Zbl1149.30033MR2366233
  26. [26] A. Zorich, Rauzy-Veech induction, Rauzy classes, generalized permutations on Mathematica, http://perso.univ-rennes1.fr/anton.zorich. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.