Near optimal spectral gaps for 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 exists a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.