Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

TL;DR

This paper establishes an upper bound on the sum of positive Lyapunov exponents for Teichmüller curves in strata of quadratic differentials with many poles, confirming a conjecture about their asymptotic behavior.

Contribution

It provides the first upper bound for Lyapunov exponents in these strata, proving Grivaux-Hubert's conjecture for cases with large zero multiplicities.

Findings

Upper bound for Lyapunov exponents established

Confirms asymptotic behavior conjecture for strata with many poles

Applicable to $SL(2, ext{R})$-invariant subspaces over $ ext{Q}$

Abstract

We show an upper bound for the sum of positive Lyapunov exponents of any Teichm\"uller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.