A coding-free simplicity criterion for the Lyapunov exponents of Teichmueller curves

TL;DR

This paper introduces a coding-free method to establish the simplicity of Lyapunov exponents for Teichmueller curves and related structures, leveraging results from harmonic analysis and boundary theory.

Contribution

It provides a novel, coding-free criterion for Lyapunov spectrum simplicity, applicable to complex Teichmueller and Calabi-Yau settings, expanding beyond traditional coding-based methods.

Findings

Proves simplicity of Lyapunov exponents for certain Teichmueller curves.

Extends simplicity results to variations of Hodge structures in Calabi-Yau threefolds.

Demonstrates the method's applicability where coding approaches are difficult.

Abstract

In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmueller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmueller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmueller curves. Finally, we extend the discussion in this note to show the simplicity of Lyapunov exponents coming from (high weight) variations of Hodge structures associated to mirror quintic Calabi-Yau threefolds.