Orbit Equivalence Rigidity for Product Actions

Drimbe, Daniel

doi:10.1007/s00220-019-03598-y

Cited by 10 publications

(3 citation statements)

References 58 publications

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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: 84%

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: 84%

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

Chifan¹,

Diaz-Arias²,

Drimbe³

2020

Preprint

Self Cite

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“…By using [DHI16, Lemma 2.4] it follows that . By applying [Dri20b, Lemma 2.3], we derive that for any subset , The rest of the proof is divided between three claims.…”

Section: Measure Equivalence and Tensor Product Decompositions For 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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“…R Λ n 1 ×...×Λ n k ≅ (R Λ m 1 ×...×Λ m l ) t . Therefore using [MS02, Theorem 1.16] (see also [Dr19,Theorem ]) we have k = l and after permuting the indices we have R Λ n i ≅ (R Λ m i ) t i for some t 1 t 2 ...t k = t. However using [Ga02, Corollaire 0.4] (see also [CZ88]) this further implies that n i = m i and t, t 1 , t 2 , ..., t k = 1; in particular, G ≅ H.…”

Section: Proofmentioning

confidence: 99%

Examples of property (T) II$_1$ factors with trivial fundamental group

Chifan¹,

Das²,

Houdayer³

et al. 2020

Preprint

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In this article we provide the first examples of property (T) II 1 factors N with trivial fundamental group, F(N ) = 1. Our examples arise as group factors N = L(G) where G belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in [Va04] and the ones introduced recently in [CDK19] using the Belegradek-Osin Rips construction from [BO06]. In particular, our results provide a continuum of explicit pairwise non-isomorphic property (T) factors.

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Orbit Equivalence Rigidity for Product Actions

Cited by 10 publications

References 58 publications

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

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

Measure equivalence rigidity via s-malleable deformations

Examples of property (T) II$_1$ factors with trivial fundamental group

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Orbit Equivalence Rigidity for Product Actions

Cited by 10 publications

References 58 publications

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

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

Measure equivalence rigidity via s-malleable deformations

Examples of property (T) II$_1$ factors with trivial fundamental group

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

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