Local spectral gap in simple Lie groups and applications

Boutonnet, Rémi; Ioana, Adrian; Golsefidy, Alireza Salehi

doi:10.1007/s00222-016-0699-8

Cited by 31 publications

(58 citation statements)

References 61 publications

(112 reference statements)

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“…This result is due to Bourgain and Gamburd for G = SO(3) [BG06], and to Benoist and de Saxcé in general [BdS14]. In joint work with Boutonnet and Salehi-Golsefidy, we have shown that the same holds if G is an arbitrary connected simple Lie group [BISG15]. In particular, the translation action Γ (G, m G ) is strongly ergodic, whenever Γ < G := SL n (R) is a dense subgroup whose elements are matrices with algebraic entries.…”

Section: Oe Rigidity For Actions Of Non-rigid Groupssupporting

confidence: 57%

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Rigidity for Von Neumann Algebras

Ioana

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W * -superrigid) actions that can be completely recovered from their von Neumann algebras and II1 factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.• the existence of non-orbit equivalent actions of non-amenable groups [GP03, Io06b, Ep07].

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Section: Oe Rigidity For Actions Of Non-rigid Groupssupporting

confidence: 57%

“…As an application, it led to a continuum of mutually non-OE and Borel incomparable actions of SL 2 (Z), confirming a conjecture from [Th01,Th06]. It also motivated a "local spectral gap" theorem for dense subgroups of simple Lie groups in [BISG15].…”

Section: Introductionmentioning

confidence: 61%

Rigidity for Von Neumann Algebras

Ioana

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…In [HI15, Section 6], for every λ ∈ (0, 1], a concrete example of a strongly ergodic free action F ∞ (X λ , µ λ ) on a standard measure space was constructed using the main result of [BISG15]. The following open question is interesting in this respect.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Strongly ergodic equivalence relations: spectral gap and type III invariants

Houdayer¹,

Marrakchi²,

Verraedt³

2017

Ergod. Th. Dynam. Sys.

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We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable 1-cocycles with values into locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and τ invariants for type III strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type III1 ergodic equivalence relation R, the Maharam extension c(R) is strongly ergodic if and only if R is strongly ergodic and the invariant τ (R) is the usual topology on R. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes' structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and τ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

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“…We finally exploit recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15] to construct, for every 0 < λ ≤ 1, examples of type III λ strongly ergodic free nonsingular actions of a free group on a standard probability space. It is shown in [BISG15, Theorem A] that for any (not necessarily compact) connected simple Lie group and any countable dense subgroup Λ < G with "algebraic entries" (e.g.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

Houdayer

Isono

2016

Commun. Math. Phys.

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Abstract. We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras M = B ⋊Γ arising from arbitrary actions Γ B of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra N ′ ∩ M ω of any nonamenable von Neumann subalgebra with normal expectation N ⊂ M . We use this result to show that for any strongly ergodic essentially free nonsingular action Γ (X, µ) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor L ∞ (X) ⋊ Γ has no nontrivial central sequence. Using recent results of Boutonnet-Ioana-Salehi Golsefidy [BISG15], we construct, for every 0 < λ ≤ 1, a type III λ strongly ergodic essentially free nonsingular action F∞ (X λ , µ λ ) of the free group F∞ on a standard probability space so that the corresponding group measure space type III λ factor L ∞ (X λ , µ λ ) ⋊ F∞ has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type III factors with no nontrivial central sequence.

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Local spectral gap in simple Lie groups and applications

Cited by 31 publications

References 61 publications

Rigidity for Von Neumann Algebras

Rigidity for Von Neumann Algebras

Strongly ergodic equivalence relations: spectral gap and type III invariants

Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence

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