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On the spectral gap of negatively curved surface covers
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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.
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Cited by 1 Pith paper
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On the spectral stability of finite coverings
Random finite covers of manifolds with Ricci curvature bounded below have no new Laplacian eigenvalues in [0,Λ] when the fundamental group satisfies strong convergence of permutation representations.
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