Commentarii Mathematici Helvetici


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Volume 78, Issue 1, 2003, pp. 116–133
DOI: 10.1007/s000140300005

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

Enrico Leuzinger[1]

(1) Karlsruhe Institute of Technology (KIT), Germany

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).

Keywords: Discrete subgroups of semisimple Lie groups, Kazhdan's property (T), isoperimetric inequalities, $L^2$ -spectrum, locally symmetric spaces, rigidity, Brownian motion

Leuzinger Enrico: Kazhdan's property (T), $L^2$-spectrum and isoperimetric inequalities for locally symmetric spaces. Comment. Math. Helv. 78 (2003), 116-133. doi: 10.1007/s000140300005