Lattices with and lattices without spectral gap

Let G = G(k) be the K-rational points of a simple algebraic group G over a local field K and let Γ be a lattice in G. We show that the regular representation ρΓ\G of G on L²(Γ\G) has a spectral gap, that is, the restriction of ρΓ\G to the orthogonal of the constants in L²(Γ\G) has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups G and lattices Γ for which L²(Γ\G) has no spectral gap. This answers in the negative a question asked by Margulis. In fact, G can be taken to be the group of orientation preserving automorphisms of a k-regular tree for K > 2.