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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