  The following two results are shown.
  1) Let $G$ be the $k$-rational points of a simple algebraic group over a
local field $k$ and let $H$ be a lattice in $G.$ Then the regular
representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no
almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with
zero mean).
  2) There exist locally compact simple groups $G$ and lattices $H$ for which
$L^2(G/H)$ 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 $k>2.$
