Prime II1 factors arising from actions of product groups

Drimbe, Daniel

doi:10.1016/j.jfa.2019.108366

Cited by 6 publications

(3 citation statements)

References 55 publications

(38 reference statements)

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“…In the first part of the section we prove Theorem A (see Theorem 5.2) and therefore, generalize the main results from [CdSS15]. The technology that we use is slightly different from the one in [CdSS15], resembling more the methods developed in [DHI16,Dr19a,Dr19b].…”

Section: W * -Superrigidity For Product Groupsmentioning

confidence: 85%

New examples of W$^$ and C$^$-superrigid groups

Chifan¹,

Diaz-Arias²,

Drimbe³

2020

Preprint

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. Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of W * -superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in C * -algebra theory including additional examples of C * -superrigid groups and explicit computations of symmetries of reduced group C * -algebras.

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Section: W * -Superrigidity For Product Groupsmentioning

confidence: 85%

New examples of W$^$ and C$^$-superrigid groups

Chifan¹,

Diaz-Arias²,

Drimbe³

2020

Preprint

Self Cite

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“…Denote . By applying [Dri20a, Lemma 2.10], we get that is strongly non-amenable relative to inside . Then Remark 3.4 implies that is -rigid.…”

Section: Malleable Deformations For Von Neumann Algebras: Classmentioning

confidence: 99%

Measure equivalence rigidity via s-malleable deformations

Drimbe¹

2023

Compositio Math.

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We single out a large class of groups ${\rm {\boldsymbol {\mathscr {M}}}}$ for which the following unique prime factorization result holds: if $\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}}$ and $\Gamma _1\times \cdots \times \Gamma _n$ is measure equivalent to a product $\Lambda _1\times \cdots \times \Lambda _m$ of infinite icc groups, then $n \ge m$ , and if $n = m$ , then, after permutation of the indices, $\Gamma _i$ is measure equivalent to $\Lambda _i$ , for all $1\leq i\leq n$ . This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Class ${\rm {\boldsymbol {\mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $\Gamma$ for which either (i) $\Gamma$ is an arbitrary wreath product group with amenable base or (ii) $\Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Finally, for groups $\Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(\Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].

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“…For proving Theorem A we need the following result, which provides sufficient conditions at the von Neumann algebra level for a pmp action to admit a non-trivial direct product decomposition (see also [Dr19,Theorem 3.1]). The factor setting is essential for the result and its proof is based on arguments from [CdSS15, Theorem 4.14] (see also [DHI16,Theorem 6.1] and [CdSS17, Theorem 4.7]).…”

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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Prime II1 factors arising from actions of product groups

Cited by 6 publications

References 55 publications

New examples of W$^$ and C$^$-superrigid groups

New examples of W$^$ and C$^$-superrigid groups

Measure equivalence rigidity via s-malleable deformations

Orbit Equivalence Rigidity for Product Actions

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Prime II1 factors arising from actions of product groups

Cited by 6 publications

References 55 publications

New examples of W$^*$ and C$^*$-superrigid groups

New examples of W$^*$ and C$^*$-superrigid groups

Measure equivalence rigidity via s-malleable deformations

Orbit Equivalence Rigidity for Product Actions

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Product

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New examples of W$^$ and C$^$-superrigid groups

New examples of W$^$ and C$^$-superrigid groups