Skip to main content

Research Repository

Advanced Search

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

Magee, Michael; Naud, Frédéric; Puder, Doron

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


Frédéric Naud

Doron Puder


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 .


Magee, M., Naud, F., & Puder, D. (2022). A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ. Geometric And Functional Analysis, 32(3), 595-661.

Journal Article Type Article
Acceptance Date Mar 15, 2022
Online Publication Date May 17, 2022
Publication Date 2022-06
Deposit Date Apr 20, 2020
Publicly Available Date Jul 14, 2022
Journal Geometric and Functional Analysis
Print ISSN 1016-443X
Electronic ISSN 1420-8970
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 32
Issue 3
Pages 595-661


Published Journal Article (1.5 Mb)

Publisher Licence URL

Copyright 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

You might also like

Downloadable Citations