Spectral gaps on thick part of moduli spaces

TL;DR

This paper investigates the behavior of spectral gaps on hyperbolic surfaces of large genus within the thick part of moduli space, revealing that the maximum gap approaches 1/4 as genus increases.

Contribution

It establishes that spectral gap differences on large genus surfaces tend to a universal limit within the thick part of moduli space.

Findings

Spectral gap differences approach 1/4 as genus increases.

Maximum spectral gap difference is achieved in the thick part of moduli space.

Results hold for any fixed number of eigenvalues k.

Abstract

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of $\lambda_k-\lambda_{k-1}$ over any thick part of the moduli space of closed Riemann surfaces approaches the limit $\frac{1}{4}$.