Lattices with and lattices without spectral gap

Abstract

Let $G=\mathbf{G}(\mathbb{k})$ be the $\mathbb{k}$-rational points of a simple algebraic group $\mathbf{G}$ over a local field $\mathbb{k}$ and let $\Gamma$ be a lattice in $G$. We show that the regular representation ${\rho }_{\Gamma \backslash G}$ of $G$ on $L^2(\Gamma \backslash G)$ has a spectral gap, that is, the restriction of ${\rho }_{\Gamma \backslash G}$ to the orthogonal of the constants in $L^2(\Gamma \backslash G)$ has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups $G$ and lattices $\Gamma$ for which $L^2(\Gamma \backslash G)$ 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 $k>2$.