Journal of Lie Theory, Vol. 10, No. 1, pp. 93-105 (2000)

On Kazhdan's property (T) and Kazhdan constants associated to a Laplacian for SL(3,R)

M. E. B. Bekka and M. Mayer

M. E. B. Bekka
Université de Metz
Ile du Saulcy
57045 Metz Cedex 01
Matthias Mayer
Zentrum Mathematik
Technische Universität München
D-80290 München

Abstract: The first purpose of this paper is to give a very elementary proof of Property (T) for SL$_{\Bbb R}(3,\kk)$ over any local field $\kk$. Here we use a modification of an argument due to Burger. The second approach to Property (T) is based on spectral properties of a Laplacian in the enveloping algebra. It is shown that for a connected Lie group $G$ Property (T) can be characterized by a spectral property of a Laplacian on the space of smooth $K$-finite vectors, where $K$ is a compact subgroup of $G$.

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