JournalsggdVol. 5, No. 2pp. 251–264

Lattices with and lattices without spectral gap

  • Bachir Bekka

    Université de Rennes I, France
  • Alexander Lubotzky

    Hebrew University, Jerusalem, Israel
Lattices with  and lattices without spectral gap cover
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Abstract

Let G=G(k)G=\boldsymbol{G}(\mathbb{k}) be the k\mathbb{k}-rational points of a simple algebraic group G\boldsymbol{G} over a local field k\mathbb{k} and let Γ\Gamma be a lattice in GG. We show that the regular representation ρΓ\G\rho_{\Gamma\backslash G} of GG on L2(Γ\G)L^2(\Gamma\backslash G) has a spectral gap, that is, the restriction of ρΓ\G\rho_{\Gamma\backslash G} to the orthogonal of the constants in L2(Γ\G)L^2(\Gamma\backslash G) has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups GG and lattices Γ\Gamma for which L2(Γ\G)L^2(\Gamma\backslash G) has no spectral gap. This answers in the negative a question asked by Margulis. In fact, GG can be taken to be the group of orientation preserving automorphisms of a kk-regular tree for k>2k>2.

Cite this article

Bachir Bekka, Alexander Lubotzky, Lattices with and lattices without spectral gap. Groups Geom. Dyn. 5 (2011), no. 2, pp. 251–264

DOI 10.4171/GGD/126