Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle

TL;DR

This paper characterizes when the Kontsevich-Zorich cocycle has zero Lyapunov exponents, linking it to geometric constraints and describing possible monodromy groups, thus clarifying the conditions for minimal zero exponents.

Contribution

It confirms a conjecture that zero Lyapunov exponents occur only under specific geometric conditions and classifies the monodromy groups involved.

Findings

Zero Lyapunov exponents occur only under certain geometric constraints.

The monodromy groups are classified as specific real Lie groups.

The number of zero exponents is minimized given the monodromy.

Abstract

We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich-Zorich cocycle. The number of zero exponents is then as small as possible, given its monodromy.