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Siegel–Veech transforms are in $ \boldsymbol{L^2} $(with an appendix by Jayadev S. Athreya and Rene Rühr)
The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces
| 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 |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
C. McMullen,
Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946.
doi: 10.1007/s00208-012-0804-2. |
| [8] |
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
C. McMullen,
Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946.
doi: 10.1007/s00208-012-0804-2. |
| [8] |
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. |
| [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. |
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