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

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 SU (p, q) type are realized by appropriate covering constructions.

Zorich conjecture for hyperelliptic Rauzy–Veech groups

“…1 See the Appendix A below for a concrete example. 2 On the other hand, Zorich conjecture implies Avila-Viana theorem on the pinching and twisting properties for Rauzy-Veech groups. In fact, Zariski density implies the pinching property by the work of Benoist [4], while the twisting property is automatic (because it has to do with minors of matrices).…”

“…Remark 1.3. In a forthcoming paper [2], we will use the framework of this article to analyze the Kontsevich-Zorich cocycle over certain loci of cyclic covers of hyperelliptic connected components of strata of the moduli space of translation surfaces. Remark 1.4.…”

“…[14,Appendix A.3]) concerning the Zariski denseness of Rauzy-Veech groups in symplectic groups. Indeed, it is known 1 among experts that some pinching and twisting groups have small Zariski closures, so that it is not possible to abstractly deduce 2 Zorich's conjecture from Avila-Viana techniques.…”

Kontsevich-Zorich monodromy groups of translation covers of some platonic solids

“…See also Matheus-Yoccoz-Zmiaikou [17] who proved constraints for these groups in the case of square-tiled surfaces. Realizability of the matrix families has been studied in [4,10,12]. Other algebraic questions concern whether the Kontsevich-Zorich monodromy groups are arithmetic (see Hubert-Matheus [14] for the existence of an arithmetic monodromy group) and how frequently this is the case Bonnafoux et.al [5].…”

A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle

“…Indeed, it is well-known that every group in the rst item is realizable. The groups in the second item were shown to be realizable by Avila-Matheus-Yoccoz [AMY17]. Moreover, the group SO * (6) in its standard representation (which coincides with SU(3, 1) in its second exterior power representation) is also realizable by the work of Filip-Forni-Matheus [FFM18].…”