Near optimal spectral gaps for hyperbolic surfaces

Hide, Will; Magee, Michael

doi:10.4007/annals.2023.198.2.6

Near optimal spectral gaps for hyperbolic surfaces

Hide, Will; Magee, Michael

Authors

William Hide william.hide@durham.ac.uk
PGR Student Doctor of Philosophy

Professor Michael Magee michael.r.magee@durham.ac.uk
Professor

Abstract

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 .

Citation

Hide, W., & Magee, M. (2023). Near optimal spectral gaps for hyperbolic surfaces. Annals of Mathematics, 198(2), 791-824. https://doi.org/10.4007/annals.2023.198.2.6

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
DOI	https://doi.org/10.4007/annals.2023.198.2.6
Public URL	https://durham-repository.worktribe.com/output/1176898