Let G=G(k) be the k-rational points of a simple algebraic group G over a local field k and let Γ be a lattice in G. We show that the regular representation ρΓ\G of G on L2(Γ\G) has a spectral gap, that is, the restriction of ρΓ\G to the orthogonal of the constants in L2(Γ\G) has no almost invariant vectors. On the other hand, we give examples of locally compact simple groups G and lattices Γ for which L2(Γ\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.
Classification :
20-XX, 00-XX
Mots-clés :
Lattices in algebraic groups, spectral gap property, automorphism groups of trees, expander diagrams
Affiliations des auteurs :
Bachir Bekka
1
;
Alexander Lubotzky
2
1
Université de Rennes I, France
2
Hebrew University, Jerusalem, Israel
Bachir Bekka; Alexander Lubotzky. Lattices with and lattices without spectral gap. Groups, geometry, and dynamics, Tome 5 (2011) no. 2, pp. 251-264. doi: 10.4171/ggd/126
@article{10_4171_ggd_126,
author = {Bachir Bekka and Alexander Lubotzky},
title = {Lattices with and lattices without spectral gap},
journal = {Groups, geometry, and dynamics},
pages = {251--264},
year = {2011},
volume = {5},
number = {2},
doi = {10.4171/ggd/126},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/126/}
}
TY - JOUR
AU - Bachir Bekka
AU - Alexander Lubotzky
TI - Lattices with and lattices without spectral gap
JO - Groups, geometry, and dynamics
PY - 2011
SP - 251
EP - 264
VL - 5
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/126/
DO - 10.4171/ggd/126
ID - 10_4171_ggd_126
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%0 Journal Article
%A Bachir Bekka
%A Alexander Lubotzky
%T Lattices with and lattices without spectral gap
%J Groups, geometry, and dynamics
%D 2011
%P 251-264
%V 5
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/126/
%R 10.4171/ggd/126
%F 10_4171_ggd_126
