On the spectral gap of negatively curved surface covers

TL;DR

This paper provides an explicit probabilistic estimate for the first non-trivial Laplacian eigenvalue on random covers of negatively curved surfaces, generalizing previous hyperbolic results and exploring near-optimal spectral gaps.

Contribution

It introduces a new explicit estimate for spectral gaps on random surface covers in variable curvature, extending prior hyperbolic surface results and proposing a conjecture on optimal gaps.

Findings

Explicit eigenvalue estimates valid with high probability as degree increases

Generalization of hyperbolic surface spectral gap results to variable curvature

Existence of covers with near-optimal spectral gaps

Abstract

Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The explicit gap is given in terms of the bottom of the spectrum of the universal cover of $X$ and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory.