Lattices with and lattices without spectral gap

The following two results are shown.

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).
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 $k > 2.$

The following two results are shown.

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).
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 $k > 2.$

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