On second eigenvalues of closed hyperbolic surfaces for large genus

TL;DR

This paper investigates the asymptotic behavior of the second eigenvalues of large genus hyperbolic surfaces, establishing bounds related to geometric features and analyzing typical ratios for random surfaces.

Contribution

It provides sharp bounds for second eigenvalues in terms of geometric length and explores the typical ratio of eigenvalues to geometric lengths for random large genus surfaces.

Findings

Second eigenvalues are bounded between constants times $rac{ ext{length}}{g^2}$ and $ ext{length}.

The ratio $rac{ ext{second eigenvalue}}{ ext{length}}$ is asymptotically comparable to $rac{1}{ ext{log}(g)}$ for random surfaces.

Bounds are shown to be optimal as genus $g$ tends to infinity.

Abstract

In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, up to uniform positive constants multiplications, the second eigenvalue $\lambda_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$; moreover these two bounds are optimal as $g\to \infty$. Here $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also investigate the quantity $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$.