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Explicit spectral gaps for random covers of Riemann surfaces

Magee, Michael; Naud, Frédéric

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Authors

Frédéric Naud



Abstract

We introduce a permutation model for random degree n covers Xn of a non-elementary convex-cocompact hyperbolic surface X = \H. Let δ be the Hausdorff dimension of the limit set of . We say that a resonance of Xn is new if it is not a resonance of X, and similarly define new eigenvalues of the Laplacian. We prove that for any > 0 and H > 0, with probability tending to 1 as n → ∞, there are no new resonances s = σ + it of Xn with σ ∈ [ 3 4 δ + ,δ] and t ∈ [−H, H]. This implies in the case of δ > 1 2 that there is an explicit interval where there are no new eigenvalues of the Laplacian on Xn. By combining these results with a deterministic ‘high frequency’ resonance-free strip result, we obtain the corollary that there is an η = η(X) such that with probability → 1 as n → ∞, there are no new resonances of Xn in the region {s : Re(s)>δ − η }.

Citation

Magee, M., & Naud, F. (2020). Explicit spectral gaps for random covers of Riemann surfaces. Publications mathématiques de l'IHÉS, 132(1), 137-179. https://doi.org/10.1007/s10240-020-00118-w

Journal Article Type Article
Acceptance Date Jun 10, 2020
Online Publication Date Jun 25, 2020
Publication Date 2020-12
Deposit Date Jul 31, 2019
Publicly Available Date Mar 29, 2024
Journal Publications mathématiques de l'IHÉS
Print ISSN 0073-8301
Electronic ISSN 1618-1913
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 132
Issue 1
Pages 137-179
DOI https://doi.org/10.1007/s10240-020-00118-w

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