Ends of locally symmetric spaces with maximal bottom spectrum

Abstract

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.