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Explicit spectral gaps for random covers of Riemann surfaces.

Magee, Michael and Naud, Frédéric (2020) 'Explicit spectral gaps for random covers of Riemann surfaces.', Publications mathématiques de l'IHÉS., 132 (1). pp. 137-179.


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 }.

Item Type:Article
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Publisher statement:This is a post-peer-review, pre-copyedit version of an article published in Publications mathématiques de l'IHÉS. The final authenticated version is available online at:
Date accepted:10 June 2020
Date deposited:21 July 2020
Date of first online publication:25 June 2020
Date first made open access:25 June 2021

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