Lattices with and lattices without spectral gap
Résumé
Let $G = G(k)$ be the $k$-rational points of a simple algebraic group G over a local field k and let $\Gamma$ be a lattice in G. We show that the regular representation $\rho_{\Gamma\setminus G}$ of $G$ on $L^{2}(\Gamma\setminus G)$ has a spectral gap, that is, the restriction of $\rho_{\Gamma\setminus G}$ to the orthogonal of the constants in $L^{2}(\Gamma\setminus G)$ has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups $G$ and lattices $\Gamma$ for which $L^{2}(\Gamma\setminus G)$ has no spectral gap. This answers in the negative a question asked by Margulis [Marg91, Chapter III, 1.12]. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a k-regular tree for $k > 2$.
Origine : Fichiers produits par l'(les) auteur(s)
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