A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$

Abstract

Let $X$ be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each $n\in \mathbf{N}$, let $X_n$ be a random degree-$n$ cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces of $X$. An eigenvalue of $X$ or $X_n$ is an eigenvalue of the associated Laplacian operator ${\Delta }_X$ or ${\Delta }_{X_n}$. We say that an eigenvalue of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove that for any $\epsilon >0$, with probability tending to 1 as $n\to \infty$, there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\epsilon$. We conjecture that the same result holds with $\frac{3}{16}$ replaced by $\frac{1}{4}$.

arXiv comment54 pages, 5 figures. Accepted for publication in GAFA. This version: journal version, incorporated referees' comments and added figures

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• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• The Asymptotic Statistics of Random Covering Surfaces
• Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps
• Core Surfaces
• Averages of determinants of Laplacians over moduli spaces for large genus
• Degenerating hyperbolic surfaces and spectral gaps for large genus