Near optimal spectral gaps for hyperbolic surfaces

TL;DR

This paper demonstrates that random covers of hyperbolic surfaces typically have spectral gaps close to the optimal value, and confirms the existence of sequences of hyperbolic surfaces with genus tending to infinity and Laplacian eigenvalues approaching the spectral bound.

Contribution

It establishes probabilistic spectral gap results for random hyperbolic surface covers and confirms the existence of large genus surfaces with eigenvalues approaching the spectral limit.

Findings

Random covers have no small eigenvalues with high probability.

Sequences of hyperbolic surfaces with eigenvalues approaching 1/4 exist.

Spectral gaps are near optimal for large genus hyperbolic surfaces.

Abstract

We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,\frac{1}{4}-\epsilon)$ other than those of $X$, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.