Explicit spectral gaps for random covers of Riemann surfaces

Magee, Michael; Naud, Frédéric

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: Jun 25, 2021
• 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