An Example of Quaternionic Kontsevich-Zorich Monodromy Group

Abstract

The features of the \(SL(2,\mathbb {R})\)-action on the moduli spaces of translation surfaces (and its applications to the study of interval exchange transformations, translation flows and billiards) are intimately related to the properties of the so-called Kontsevich-Zorich (KZ) cocycle. In particular, it is not surprising that the KZ cocycle is one of the main actors in the recent groundbreaking work of Eskin and Mirzakhani spsciteEsMi towards the classification of \(SL(2,\mathbb {R})\)-invariant measures on moduli spaces of translation surfaces.