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A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ

Magee, Michael and Naud, Frédéric and Puder, Doron (2022) 'A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ.', Geometric and Functional Analysis, 32 (3). pp. 595-661.

Abstract

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature −1. For each n ∈ N, let Xn be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or Xn is an eigenvalue of the associated Laplacian operator ΔX or ΔXn. We say that an eigenvalue of Xn is new if it occurs with greater multiplicity than in X. We prove that for any ε > 0, with probability tending to 1 as n → ∞, there are no new eigenvalues of Xn below 3 16 − ε. We conjecture that the same result holds with 3 16 replaced by 1 4 .

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s00039-022-00602-x
Publisher statement:This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Date accepted:15 March 2022
Date deposited:14 July 2022
Date of first online publication:17 May 2022
Date first made open access:14 July 2022

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