Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≦λ1(X). We show that if equality λ 1(⌈\X) = λ1(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 ℝ1×N endowed with a multi-warped metric, where N is compact.
Bibliographical noteFunding Information:
The first author was partially supported by NSF grant DMS-0604878. The second author was partially supported by NSF grant DMS-0503735. The third author was partially supported by NSF grant DMS-0706706.