W*-rigidity for the von Neumann algebras of products of hyperbolic groups

Chifan, Ionuţ; Santiago, Rolando de; Sinclair, Thomas

doi:10.1007/s00039-016-0361-z

Cited by 20 publications

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“…The result is sharp and complements the previous finite product rigidity property found in [CdSS16]. Using this we provide an uncountable family of restricted wreath products Γ ∼ = Σ ≀ ∆ of icc, property (T) groups Σ, ∆ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras L(Γ).…”

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confidence: 75%

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We show that any infinite collection (Γ n ) n∈N of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic infinite product rigidity phenomenon. If Λ is an arbitrary group such that L(⊕ n∈N Γ n ) ∼ = L(Λ) then there exists an infinite directThe result is sharp and complements the previous finite product rigidity property found in [CdSS16]. Using this we provide an uncountable family of restricted wreath products Γ ∼ = Σ ≀ ∆ of icc, property (T) groups Σ, ∆ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras L(Γ).

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confidence: 74%

“…Now consider Ω := vC Λ (Λ i ) = {λ ∈ Λ | |λ Λ i | < ∞}, the virtual centralizer of Λ i in Λ. Using [CdSS16,Claim 4.7] we have [Λ : Λ i Ω] < ∞ and hence Λ i Ω Λ is an icc subgroup; in particular, vZ(Λ i Ω) = 1. Consider vZ(Ω) = {ω ∈ Ω | |ω Ω | < ∞}, the virtual center of Ω.…”

Section: Proof Of Claim 37 Since D ⊆ Ql(λ I )Q Is a Finite Index Inmentioning

confidence: 99%

“…Isono studied unique prime factorization aspects for infinite tensor product of factors and several interesting results have emerged in [Is16]. Motivated in part by these results it is natural to investigate whether "product rigidity" properties, similar with ones in [CdSS16], would hold in the context of infinite direct sums groups. Specifically, if one considers Γ = ⊕ n∈N Γ n with Γ n 's icc non-amenable groups, it would be interesting to understand how much of the infinite direct sum structure of Γ is retained by its von Neumann algebra L(Γ).…”

Section: Introductionmentioning

confidence: 99%

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Some rigidity results for II1 factors arising from wreath products of property (T) groups

Chifan

Udrea

2020

Journal of Functional Analysis

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confidence: 74%

Section: Proof Of Claim 37 Since D ⊆ Ql(λ I )Q Is a Finite Index Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some rigidity results for II1 factors arising from wreath products of property (T) groups

Chifan

Udrea

2020

Journal of Functional Analysis

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“…Following [CdSS15], we denote by O Σ (g) = {hgh −1 |h ∈ Σ} the orbit of g ∈ Λ under the conjugation action of Σ. Note that…”

Section: From Tensor Decompositions To Decompositions Of Actionsmentioning

confidence: 99%

Orbit Equivalence Rigidity for Product Actions

Drimbe

2019

Commun. Math. Phys.

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Let Γ1, . . . , Γn be hyperbolic, property (T) groups, for some n ≥ 1. We prove that if a product Γ1 × · · · × Γn X1 × · · · × Xn of measure preserving actions is stably orbit equivalent to a measure preserving action Λ Y , then there exists a direct product decomposition Λ = Λ1×· · ·×Λn into n infinite groups. Moreover, there exists a measure preserving action ΛiYi which is stably orbit equivalent to Γi Xi, for any 1 ≤ i ≤ n, and the product action Λ1 × · · · × Λn Y1 × · · · × Yn is isomorphic to Λ Y .

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Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

Chifan

Santiago

Sucpikarnon

2018

Commun. Math. Phys.

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In this paper we introduce a new family of icc groups Γ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II 1 factor L(Γ) arise only from the canonical direct product decompositions of the underlying group Γ. Our groups are assembled from certain HNNextensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence we obtain several new examples of groups that give rise to prime factors.of the corresponding II 1 factor L(Γ) can arise only from the canonical direct product decompositions of the underlying group Γ. Pant and the second author showed the same result holds when Γ is a poly-hyperbolic group with non-amenable factors in its composition series [dSP17]. In this paper we introduce several new classes of groups for which this tensor product rigidity phenomenon still holds. Our groups arise from natural algebraic constructions involving HNN-extensions or amalgamated free products thus including many remarkable groups intensively studied throughout mathematics such as graph products or poly-amalgams groups.Basic properties in Bass-Serre theory of groups show that the only way an amalgam Γ 1 * Σ Γ 2 could decompose as a direct product is through its core Σ.An interesting question is to investigate situations when this basic group theoretic aspect could be upgraded to the von Neumann algebraic setting. It is known this fails in general since there are examples of product indecomposable icc amalgams whose corresponding factors are McDuff and hence decomposable as tensor products (see Remarks 6.3). However, under certain indecomposability assumptions on the core algebra (see also Remarks 6.3) we are able to provide a positive answer to our question.Theorem A. Let Γ = Γ 1 * Σ Γ 2 be an icc group such that [Γ 1 : Σ] ≥ 2 and [Γ 2 : Σ] ≥ 3. Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L2 , for some groups Σ 0 Γ 0 1 , Γ 0 2 , and hence Γ = Ω × (Γ 0 1 * Σ 0 Γ 0 2 ). Moreover, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(Γ 0 1 * Σ 0 Γ 0 2 ) 1/t u * .(1.1)The same question can be considered in the realm of non-degenerate HNN-extension groups and a similar approach leads to the following counterpart of Theorem A. Theorem B. Let Γ = HNN(Λ, Σ, θ) be an icc group such that Σ = Λ = θ(Σ). Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L(Γ) = M 1⊗ M 2 , for diffuse M i 's. Then there exist decompositions Σ = Ω × Σ 0 with Σ 0 finite and Λ = Ω × Λ 0 . In addition, there is ω ∈ Ω such that θ = ad(ω) |Ω × θ |Σ 0 : Ω × Σ 0 → Ω × Λ 0 and hence Γ = Ω × HNN(Λ 0 , Σ 0 , θ |Σ 0 ). Also, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(HNN(Λ 0 , ...

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W*-rigidity for the von Neumann algebras of products of hyperbolic groups

Cited by 20 publications

References 29 publications

Some rigidity results for II1 factors arising from wreath products of property (T) groups

Some rigidity results for II1 factors arising from wreath products of property (T) groups

Orbit Equivalence Rigidity for Product Actions

Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

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