Piecewise strongly proximal actions, free boundaries and the Neretin groups

Caprace, Pierre–Emmanuel; Boudec, Adrien Le; Bon, Nicolás Matte

doi:10.48550/arxiv.2107.07765

Cited by 3 publications

(3 citation statements)

References 23 publications

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“…Let 𝐹 be a nonabelian finite simple group. Then the semirestricted wreath product A few comments relevant to Conjecture 6.1 may be found in the introduction of [23]. Known results in the discrete case make it natural to strengthen that conjecture as follows.…”

Section: Type I Hyperbolic Groupsmentioning

confidence: 95%

A type I conjecture and boundary representations of hyperbolic groups

Caprace

Kalantar

Monod

2023

Proceedings of London Math Soc

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We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group 𝐺 associated with nonsingular 𝐺-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that nonamenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

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Section: Type I Hyperbolic Groupsmentioning

confidence: 95%

A type I conjecture and boundary representations of hyperbolic groups

Caprace

Kalantar

Monod

2023

Proceedings of London Math Soc

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“…If some point of the Furstenberg boundary of G has a trivial stabilizer, then C * r (G) is simple. A few comments relevant to Conjecture 6.1 may be found in the introduction of [CBB21].…”

Section: Conjectures and Relation To C * -Simplicitymentioning

confidence: 99%

A type I conjecture and boundary representations of hyperbolic groups

Caprace¹,

Kalantar²,

Monod³

2021

Preprint

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We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group G associated with non-singular G-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of L. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

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“…Recently, P.-E. Caprace, A. Le Boudec and N. Matte Bon proved that the Neretin group N d,k is not of type I by constructing two weakly equivalent but inequivalent irreducible representations of N d,k [4]. In their paper, they conjectured that the subgroup O d,k of the Neretin group N d,k is not type I either [4,Remark 4.8].…”

Section: Introductionmentioning

confidence: 99%

On the type of the von Neumann algebra of an open subgroup of the Neretin group

Arimoto

2022

Proc. Amer. Math. Soc. Ser. B

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The Neretin group N d , k \mathcal {N}_{d, k} is the totally disconnected locally compact group consisting of almost automorphisms of the tree T d , k \mathcal {T}_{d, k} . This group has a distinguished open subgroup O d , k \mathcal {O}_{d, k} . We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether O d , k \mathcal {O}_{d, k} is of type I or not.

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Piecewise strongly proximal actions, free boundaries and the Neretin groups

Cited by 3 publications

References 23 publications

A type I conjecture and boundary representations of hyperbolic groups

A type I conjecture and boundary representations of hyperbolic groups

A type I conjecture and boundary representations of hyperbolic groups

On the type of the von Neumann algebra of an open subgroup of the Neretin group

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