A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle

TL;DR

This paper establishes a geometric criterion for non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on moduli spaces, with broad applications to various invariant measures and surfaces.

Contribution

It introduces a new geometric criterion that simplifies and generalizes previous proofs of non-uniform hyperbolicity for the Kontsevich--Zorich cocycle.

Findings

Criterion applies to measures on all algebraically primitive Veech surfaces

Criterion applies to Prym eigenforms and canonical measures

Simplifies previous proofs of hyperbolicity

Abstract

We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $\SL$-orbits of all algebraically primitive Veech surfaces (see also \cite{Bouw:Moeller}) and of all Prym eigenforms discovered in \cite{McMullen2}, as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also \cite{Ftwo}, \cite{Avila:Viana}). The argument simplifies and generalizes our proof for the case of canonical measures \cite{Ftwo}. In an Appendix Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.