A spectral gap property for subgroups of finite covolume in Lie groups

Bekka, Bachir; Cornulier, Yves de

doi:10.4064/cm118-1-9

Cited by 7 publications

(8 citation statements)

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“…In this paper, Γ is a discrete group, K is a (complex) Hilbert space and π is a unitary representation of Γ on K. We denote by 1 Γ the one dimensional trivial representation of Γ, and K π the set of πinvariant vectors in K. Recall that (see [2]) π is said to have a spectral gap if the restriction of π on the orthogonal complement (K π ) ⊥ does not weakly contain 1 Γ (in the sense of Fell), i.e. there does not exist a net of unit vectors ξ i ∈ (K π ) ⊥ satisfying π t ξ i − ξ i → 0 for every t ∈ Γ.…”

Section: Introductionmentioning

confidence: 99%

Spectral Gap Actions and Invariant States

2013

International Mathematics Research Notices

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We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group Γ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra L(Γ). Moreover, a countable discrete group Γ has property (T ) if and only if for any action α of Γ on a von Neumann algebra N , every α-invariant state on N is a weak- * -limit of a net of normal α-invariant states.

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Section: Introductionmentioning

confidence: 99%

Spectral Gap Actions and Invariant States

2013

International Mathematics Research Notices

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“…In a first version of this paper, I had sketched a proof, similar to [18, §VII.1], relying on Margulis' arithmeticity theorem and the Harish-Chandra-Borel-Behr-Harder reduction theorem for S-arithmetic lattices. François Maucourant explained to me that this is a direct consequence of property (T ) (actually even of spectral gap), and that it applies more generally to all Lie groups and all simple algebraic groups over local fields [2], [5], [4]. I thank him for allowing me to include this proof here.…”

Section: Facts On Latticesmentioning

confidence: 93%

Strong property (T) for higher rank lattices

Salle

2019

Acta Mathematica

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la salle in) semisimple groups rather than simple groups, and also to some non semisimple Lie groups (Remark 4.3).In the whole article, local field will mean commutative, non-discrete locally compact topological field. So a local field is a finite extension of R (in which case it is Archimedean), or of Q p or F p ((t)) for some prime number p (in which case it is non-Archimedean). Higher-rank simple group will mean either real connected simple Lie group of real rank 2, or connected almost F-simple algebraic group of F-split rank 2 over a local field F. Higher-rank group will stand for a finite product of higherrank simple groups. We warn the reader that for us, products of rank-1 groups such as SL 2 (R)×SL 2 (Q p ) are not of higher rank. We refer to [18, Chapter I] for the terminology. Note that real connected simple Lie group or real rank 2 is more general than connected almost simple algebraic group of split rank 2 over R. It includes for example some groups with infinite center, as the infinite covering group of Sp 2n (R).Recall that a lattice in a locally compact group G is a discrete subgroup Γ such that G/Γ carries a G-invariant Borel probability measure.Theorem 1.1. Every lattice in a higher-rank group has strong property (T ).Examples of lattices in higher-rank groups include SL n (Z), SL n (F p [t]), SL n (Z[1/p]), for n 3, and Sp 2n (Z), Sp 2n (Z) (the preimage of Sp 2n (Z) in the universal cover of Sp 2n (R)), Sp 2n (F p [X]), for n 2. None of these examples is a cocompact lattice, so for all these cases Theorem 1.1 is new.

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“…Part (i) of the following theorem has been conjectured in [Marg91, Chapter III. Remark 1.12] and proved in [BeCo08]; part (ii) is from [BeLu11]. Concerning part (ii) of the theorem, observe that when k is non-archimedean with characteristic 0, every lattice Γ in G(k) is cocompact (see [Serr,p.84]) and the result follows from Proposition 8.1.…”

Section: The Case Of a Non Cocompact Latticementioning

confidence: 98%

Spectral rigidity of group actions on homogeneous spaces

Bekka

2016

Preprint

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Actions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striking applications to group theory, geometry, ergodic theory, operator algebras, graph theory, theoretical computer science, etc.

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A spectral gap property for subgroups of finite covolume in Lie groups

Abstract: Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G.

Cited by 7 publications

References 8 publications

Spectral Gap Actions and Invariant States

Spectral Gap Actions and Invariant States

Strong property (T) for higher rank lattices

Spectral rigidity of group actions on homogeneous spaces

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