Let be the -rational points of a simple algebraic group over a local field and let be a lattice in . We show that the regular representation of on has a spectral gap, that is, the restriction of to the orthogonal of the constants in has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups and lattices for which has no spectral gap. This answers in the negative a question asked by Margulis. In fact, can be taken to be the group of orientation preserving automorphisms of a -regular tree for .
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Bachir Bekka, Alexander Lubotzky, Lattices with and lattices without spectral gap. Groups Geom. Dyn. 5 (2011), no. 2, pp. 251–264DOI 10.4171/GGD/126