# Groups, Geometry, and Dynamics

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**Volume 5, Issue 2, 2011, pp. 251–264**

**DOI: 10.4171/GGD/126**

Published online: 2011-03-06

Lattices with and lattices without spectral gap

Bachir Bekka^{[1]}and Alexander Lubotzky

^{[2]}(1) Université de Rennes I, France

(2) Hebrew University, Jerusalem, Israel

Let $G=\boldsymbol{G}(\mathbb{k})$ be the $\mathbb{k}$-rational points of a simple algebraic group $\boldsymbol{G}$ 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 $k>2$.

*Keywords: *Lattices in algebraic groups, spectral gap property, automorphism groups of trees, expander diagrams

Bekka Bachir, Lubotzky Alexander: Lattices with and lattices without spectral gap. *Groups Geom. Dyn.* 5 (2011), 251-264. doi: 10.4171/GGD/126