Advertisement

Mathematische Annalen

, Volume 370, Issue 1–2, pp 785–809 | Cite as

Zorich conjecture for hyperelliptic Rauzy–Veech groups

  • Artur Avila
  • Carlos MatheusEmail author
  • Jean-Christophe Yoccoz
Article

Abstract

We describe the structure of hyperelliptic Rauzy diagrams and hyperelliptic Rauzy–Veech groups. In particular, this provides a solution of the hyperelliptic cases of a conjecture of Zorich on the Zariski closure of Rauzy–Veech groups.

Notes

Acknowledgements

The authors are thankful to Pascal Hubert and Martin Möller for pointing out to us the references [1, 11]. Also, the authors are grateful to the two referees for their careful reading of this text.

References

  1. 1.
    A’Campo, N.: Tresses, monodromie et le groupe symplectique. Comment. Math. Helv. 54(2), 318–327 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avila, A., Matheus, C., Yoccoz, J.-C.: On the Kontsevich-Zorich cocycle for the Veech–McMullen family of symmetric translation surfaces (in preparation)Google Scholar
  3. 3.
    Avila, A., Viana, M.: Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture. Acta Math. 198, 1–56 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benoist, Y.: Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7, 1–47 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eskin, A., Filip, S., Wright, A.: The algebraic hull of the Kontsevich–Zorich cocycle (preprint) (2017). arXiv:1702.02074
  6. 6.
    Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012). ISBN: 978-0-691-14794-9Google Scholar
  7. 7.
    Filip, S.: Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle. Duke Math. J. 166(4), 657–706 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Forni, G.: Deviation of Ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. 155(1), 1–103 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matheus, C., Möller, M., Yoccoz, J.-C.: A criterion for the simplicity of the Lyapunov exponents of square-tiled surfaces. Invent. Math. 202(1), 333–425 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Looijenga, E., Mondello, G.: The fine structure of the moduli space of abelian differentials in genus 3. Geom. Dedicata 169, 109–128 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rauzy, G.: Échanges d’intervalles et transformations induites. Acta Arith. 34(4), 315–328 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Witte-Morris, D.: Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2005)zbMATHGoogle Scholar
  13. 13.
    Yoccoz, J.-C.: Interval exchange maps and translation surfaces. Homog. Flows Moduli Spaces Arith. 10, 1–69, Clay Math. Proc. (2010)Google Scholar
  14. 14.
    Zorich, A.: How do the leaves of a closed 1-form wind around a surface? Pseudoperiodic topology, 135–178. American Mathematical Society Translational Series, vol. 2, p. 197. American Mathematical Society, Providence (1999)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Artur Avila
    • 1
    • 2
  • Carlos Matheus
    • 3
    Email author
  • Jean-Christophe Yoccoz
    • 4
  1. 1.CNRS UMR 7586, Institut de Mathématiques de Jussieu - Paris Rive GaucheBâtiment Sophie GermainParis Cedex 13France
  2. 2.IMPARio de JaneiroBrazil
  3. 3.CNRS (UMR 7539)Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance
  4. 4.Collège de France (PSL)ParisFrance

Personalised recommendations