Affine Manifolds and Zero Lyapunov Exponents in Genus 3

TL;DR

This paper proves the non-existence of higher-dimensional orbit closures with zero Lyapunov exponents in genus three and classifies certain Teichmüller curves with specific Lyapunov properties, advancing understanding of the dynamics in this setting.

Contribution

It establishes the non-existence of higher-dimensional orbit closures with zero Lyapunov exponents in genus three and classifies Teichmüller curves with two zero exponents.

Findings

No higher-dimensional orbit closures with zero Lyapunov exponents in genus three.

Teichmüller curves with two zero exponents are in the principal stratum.

Finitely many such Teichmüller curves exist.

Abstract

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichm\"uller curve in genus three has two zero Lyapunov exponents in the Kontsevich-Zorich cocycle, then it lies in the principal stratum and has at most quadratic trace field. Moreover, there can be at most finitely many such Teichm\"uller curves.