On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials

TL;DR

This paper extends Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials, providing an approximation that generalizes the classical theorem to higher genus surfaces.

Contribution

We prove an approximation to Yoccoz's extension of Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials, a significant step beyond previous results.

Findings

Approximate validation of Yoccoz's conjecture

Generalization of Selberg's 3/16 theorem to higher genus

Insights into spectral properties of moduli spaces

Abstract

J.-C. Yoccoz proposed a natural extension of Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg's $\frac{3}{16}$ Theorem to moduli spaces of abelian differentials on surfaces of genus $\geq 2$.