𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

Kyed, David; Petersen, Henrik Densing; Vaes, Stefaan

doi:10.1090/s0002-9947-2015-06449-6

Cited by 33 publications

(52 citation statements)

References 32 publications

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“…Any reasonable definition of cost for non-discrete locally compact groups would thus only provide three distinct orbit equivalence classes: the ones with a corresponding R Γ of cost 1, the ones with R Γ of finite cost greater than 1, and the ones with R Γ of infinite cost. This mirrors the first L 2 Betti number of locally compact unimodular groups, see [Pet13] and [KPV15].…”

Section: Weak Orbit Equivalence Versus Orbit Equivalencesupporting

confidence: 62%

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Orbit full groups for locally compact groups

Carderi

Maître

2017

Trans. Amer. Math. Soc.

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We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group.We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

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Section: Weak Orbit Equivalence Versus Orbit Equivalencesupporting

confidence: 62%

“…Proof. In this proof we will use the notations and conventions of Proposition 4.3 of [KPV15]. Let Y ⊂ X be a cross section and let U ⊂ G be a neighborhood of the identity as in Definition 1.14.…”

Section: Cross-sections and Product Decompositionmentioning

confidence: 99%

Orbit full groups for locally compact groups

Carderi

Maître

2017

Trans. Amer. Math. Soc.

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“…It follows from [KPV13,Proposition 3.1] that this definition coincides with Gaboriau's definition of L 2 -Betti numbers, [G01].…”

Section: Cohomology Of Cartan Subalgebrasmentioning

confidence: 96%

“…Here, we use Lück's dimension function for arbitrary modules over a von Neumann algebra with a semifinite trace, see [L02,Section 6.1] and [KPV13,Section A.4]. …”

Section: Cohomology Of Quasi-regular Inclusions Of Von Neumann Algebrasmentioning

confidence: 99%

Cohomology and $L^2$-Betti Numbers for Subfactors and Quasi-Regular Inclusions

Popa¹,

Shlyakhtenko²,

Vaes

2017

Int Math Res Notices

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We introduce L 2 -Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II 1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups Γ, we recover the ordinary (co)homology of Γ. For Cartan subalgebras, we recover Gaboriau's L 2 -Betti numbers for the associated equivalence relation. In this common framework, we prove that the L 2 -Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the L 2 -Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.

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“…Hence Γ ′ cannot be a lattice in a product of two non-compact lcsc groups. That is because β (2) 1 (Γ ′ ) > 0 implies that the first ℓ 2 -Betti number of any lattice envelope of Γ ′ is positive [42,Theorem B], and the first ℓ 2 -Betti number of a product of two non-compact unimodular lcsc groups is zero [54, Theorem 6.5 on p. 61]. This excludes that Γ ′ < G ′ is an Sarithmetic lattice as in Setup 4.1, so type (2) is ruled out.…”

Section: Proof Of Theorem Dmentioning

confidence: 99%

Lattice envelopes

2020

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We introduce a class of countable groups by some abstract grouptheoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ 2 -Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

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𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

Cited by 33 publications

References 32 publications

Orbit full groups for locally compact groups

Orbit full groups for locally compact groups

Cohomology and $L^2$-Betti Numbers for Subfactors and Quasi-Regular Inclusions

Lattice envelopes

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