On the spectral gap of negatively curved surface covers

2025/02/15 by Will Hide, Hide, Will, Julien Moy +3 · 2 citations
Engineering · #35P15 #37D40 #60B20 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2502.10733

openalex publication_date 2025/02/15 · openalex created_date 2025/02/19 · openalex updated_date 2026/07/28

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.

Cited by

• Strong convergence: a short survey
• On the spectral stability of finite coverings

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