Near optimal spectral gaps for hyperbolic surfaces

Hide, Will; Magee, Michael

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
• Electronic ISSN: 1939-8980
• 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