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Near optimal spectral gaps for hyperbolic surfaces

Hide, Will; Magee, Michael

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William Hide
PGR Student Doctor of Philosophy


We prove that if X is a finite area non-compact hyperbolic surface, then for any ϵ > 0, with probability tending to one as n → ∞, a uniformly random degree n Riemannian cover of X has no eigenvalues of the Laplacian in [0, 1 4 − ϵ) other than those of X, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exists a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to 1 4 .


Hide, W., & Magee, M. (2023). Near optimal spectral gaps for hyperbolic surfaces. Annals of Mathematics, 198(2), 791-824.

Journal Article Type Article
Acceptance Date Feb 14, 2023
Online Publication Date Aug 31, 2023
Publication Date 2023
Deposit Date Mar 30, 2023
Publicly Available Date Aug 31, 2023
Journal Annals of Mathematics
Print ISSN 0003-486X
Publisher Department of Mathematics
Peer Reviewed Peer Reviewed
Volume 198
Issue 2
Pages 791-824
Public URL


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