Amenable invariant random subgroups

Bader, Uri; Duchesne, Bruno; Lécureux, Jean; Wesolek, Phillip

doi:10.1007/s11856-016-1324-7

Cited by 22 publications

(33 citation statements)

References 19 publications

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“…The notion of an invariant random subgroup was introduced in [1], and the problem was raised whether every amenable invariant random subgroup is concentrated on the amenable radical. This problem was recently solved affirmatively in [5]. Combining Theorem 4.1 and [59, Corollary 5.15], we obtain a different proof.…”

Section: Uniqueness Of the Trace And Amenable Irsmentioning

confidence: 64%

“…The proofs of these results occupy the first three sections of this paper, and are remarkably short and self-contained. The last result is also used (in combination with an observation from [59,Theorem 5.14]) to obtain another proof of the fact, recently proved in [5], that amenable invariant random subgroups of a discrete group concentrate on the amenable radical.…”

Section: Introductionmentioning

confidence: 93%

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C*-simplicity and the unique trace property for discrete groups

et al. 2017

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A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.Comment: 40 pages; major restructuring; four new sections added, including a new characterization of C*-simplicity, a new proof of the Kalantar-Kennedy characterization of C*-simplicity and a discussion of the Connes-Sullivan propert

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Section: Uniqueness Of the Trace And Amenable Irsmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 93%

C*-simplicity and the unique trace property for discrete groups

et al. 2017

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“…The study of G-invariant Borel probability measures on Sub(G), named invariant random subgroups (IRS's) after [AGV14], is a fast-developing topic [AGV14, ABB + 12, Gla14,BDL16]. In this paper we are interested in their topological counterparts, called uniformly recurrent subgroups (URS's) [GW15].…”

Section: Introductionmentioning

confidence: 99%

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

Boudec¹,

Bon²

2018

Ann. Sci. École Norm. Sup.

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We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a C * -simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group V is C * -simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented C * -simple groups without free subgroups. We prove that a branch group is either amenable or C * -simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group F is non-amenable, then Thompson's group T must be C * -simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups F , T and V , for which we also deduce rigidity results for their minimal actions on compact spaces.

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“…The following fundamental fact is essentially due to H. Furstenberg; see for example [10,Lemma 16.7]. For completeness, we give a proof via the main result of [2].…”

Section: 2mentioning

confidence: 95%

Indicability, residual finiteness, and simple subquotients of groups acting on trees

Caprace

Wesolek

2018

Geom. Topol.

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We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto Z. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups. arXiv:1708.04590v1 [math.GR]

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Amenable invariant random subgroups

Cited by 22 publications

References 19 publications

C*-simplicity and the unique trace property for discrete groups

C*-simplicity and the unique trace property for discrete groups

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

Indicability, residual finiteness, and simple subquotients of groups acting on trees

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