Abstract

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 \begin{document}$ SU(p,q) $\end{document} type are realized by appropriate covering constructions.

Highlights

  • We describe the Kontsevich–Zorich cocycle over an affine invariant orbifold coming from a covering construction inspired by works of Veech and McMullen

  • 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

  • We hope that the arguments in this paper might be useful to study the question of non-continuity of the central Oseledets subspaces of the Kontsevich–Zorich cocycle

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Summary

THE VEECH–MCMULLEN FAMILY OF SYMMETRIC TRANSLATION SURFACES

The image in M1,l of the ramification set of the natural projection Mk,l → M1,l is invariant under the group of affine homeomorphisms of M1,l. It follows from a result of Gutkin–Judge [4] that Mk,l is a Veech surface. The matrices in RV (k, l) preserve the natural symplectic intersection form on the absolute homology of the translation surfaces in Fk,l. We hope that the arguments in this paper might be useful to study the question of non-continuity of the central Oseledets subspaces of the Kontsevich–Zorich cocycle

HOMOLOGY GROUPS
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