Kazhdan's property (T), $L^2$-spectrum and isoperimetric inequalities for locally symmetric spaces

Abstract

Let $V=\Gamma \backslash G/K$ be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups $\Gamma \subset G$ in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).