Temperedness of L 2 (Γ\G) and positive eigenfunctions in higher rank

Edwards, Sam; Oh, Hee

Abstract

Let G = SO • (n, 1) × SO • (n, 1) and X = H n × H n for n ≥ 2. For a pair (π1, π2) of non-elementary convex cocompact representations of a finitely generated group Σ into SO • (n, 1), let Γ = (π1 × π2)(Σ). Denoting the bottom of the L 2-spectrum of the negative Laplacian on Γ\X by λ0, we show: (1) L 2 (Γ\G) is tempered and λ0 = 1 2 (n − 1) 2 ; (2) There exists no positive Laplace eigenfunction in L 2 (Γ\X). In fact, analogues of (1)-(2) hold for any Anosov subgroup Γ in the product of at least two simple algebraic groups of rank one as well as for Hitchin subgroups Γ < PSL d (R), d ≥ 3. Moreover, if G is a semisimple real algebraic group of rank at least 2, then (2) holds for any Anosov subgroup Γ of G.

Citation

Edwards, S., & Oh, H. (2023). Temperedness of L 2 (Γ\G) and positive eigenfunctions in higher rank. Communications of the American Mathematical Society, 3, 744-778. https://doi.org/10.1090/cams/25

• Journal Article Type: Article
• Acceptance Date: Aug 23, 2023
• Online Publication Date: Sep 26, 2023
• Publication Date: 2023-09
• Deposit Date: Sep 22, 2023
• Publicly Available Date: May 22, 2024
• Journal: Communications of the American Mathematical Society
• Print ISSN: 2692-3688
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 3
• Pages: 744-778
• DOI: https://doi.org/10.1090/cams/25
• Public URL: https://durham-repository.worktribe.com/output/1746133
• Publisher URL: https://www.ams.org/publications/journals/journalsframework/aboutcams