Quadratic differentials in low genus: exceptional and non-varying

TL;DR

This paper classifies components of exceptional quadratic differential strata in low genus and demonstrates that many such strata have invariant Lyapunov exponents across Teichmueller curves, revealing uniform dynamical properties.

Contribution

It provides an algebraic classification of exceptional quadratic differential strata in genus three and four, and shows the invariance of Lyapunov exponents within these strata.

Findings

Complete list of exceptional strata in genus three and four.

Many strata have constant sum of Lyapunov exponents for all Teichmueller curves.

Connection between strata components and disjointness of Teichmueller curves with Brill-Noether divisors.

Abstract

We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general investigation of disjointness of Teichmueller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum.