TY - JOUR
T1 - Lattices with and lattices without spectral gap
AU - Bekka, Bachir
AU - Lubotzky, Alexander
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
KW - Automorphism groups of trees
KW - Expander diagrams
KW - Lattices in algebraic groups
KW - Spectral gap property
UR - http://www.scopus.com/inward/record.url?scp=79952483596&partnerID=8YFLogxK
U2 - 10.4171/GGD/126
DO - 10.4171/GGD/126
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AN - SCOPUS:79952483596
SN - 1661-7207
VL - 5
SP - 251
EP - 264
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
IS - 2
ER -