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  2019, 14: 21-54. doi: 10.3934/jmd.2019002

The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

Artur Avila 1, , Carlos Matheus 2, and  Jean-Christophe Yoccoz 3,

1. 

IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
2. 

Centre de Mathématiques Laurent Schwartz, CNRS (UMR 7640), École Polytechnique, 91128 Palaiseau, France
3. 

Collège de France, 3 Rue d'Ulm, Paris, Cedex 05, France

Received  December 20, 2017 Revised  July 12, 2018 Published  March 2019

We describe the Kontsevich–Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich–Zorich monodromies of $ SU(p,q) $ type are realized by appropriate covering constructions.

Keywords: Teichmüller dynamics, cyclic covers, Veech-McMullen families, hyperelliptic Rauzy diagrams, complex reflection group.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37C85.
Citation: Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
References:
[1]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy–Veech groups, Math. Ann., 370 (2018), 785-809.  doi: 10.1007/s00208-017-1568-5.  Google Scholar

[2]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich–Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[3]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706.  doi: 10.1215/00127094-3715806.  Google Scholar

[4]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[5]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  doi: 10.1007/s00222-003-0303-x.  Google Scholar

[6]

C. Matheus, M. Möller and J.-C. Yoccoz, A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces, Invent. Math., 202 (2015), 333-425.  doi: 10.1007/s00222-014-0565-5.  Google Scholar

[7]

C. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946.  doi: 10.1007/s00208-012-0804-2.  Google Scholar

[8]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[9]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

show all references

References:
[1]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy–Veech groups, Math. Ann., 370 (2018), 785-809.  doi: 10.1007/s00208-017-1568-5.  Google Scholar

[2]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich–Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.  Google Scholar

[3]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706.  doi: 10.1215/00127094-3715806.  Google Scholar

[4]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[5]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.  doi: 10.1007/s00222-003-0303-x.  Google Scholar

[6]

C. Matheus, M. Möller and J.-C. Yoccoz, A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces, Invent. Math., 202 (2015), 333-425.  doi: 10.1007/s00222-014-0565-5.  Google Scholar

[7]

C. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946.  doi: 10.1007/s00208-012-0804-2.  Google Scholar

[8]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[9]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

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