Ends of locally symmetric spaces with maximal bottom spectrum

Lizhen Ji, Peter Li, Jiaping Wang
2009 Journal für die Reine und Angewandte Mathematik  
Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ 1 (Γ\X) satisfies the inequality λ 1 (Γ\X) ≤ λ 1 (X). We show that if equality λ 1 (Γ\X) = λ 1 (X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to R 1 × N endowed with a multi-warped metric, where N is compact.
doi:10.1515/crelle.2009.048 fatcat:ena7zx3lsbdw7n7xe4qfco6dfu