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
bekka@poncelet.univ-metz.fr
and
Matthias Mayer
Zentrum Mathematik
Technische Universität München
D-80290 München
mayerm@mathematik.tu-muenchen.de

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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