The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Adrian Ioana - One of the best experts on this subject based on the ideXlab platform.
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A CLASS OF SUPERRIGID Group VON NEUMANN ALGEBRAS
Annals of Mathematics, 2013Co-Authors: Adrian Ioana, Sorin Popa, Stefaan VaesAbstract:We prove that for any Group G in a fairly large class of generalized wreath product Groups, the associated von Neumann algebra LG completely \remembers" the Group G. More precisely, if LG is isomorphic to the von Neumann algebra L of an arbitrary Countable Group , then must be isomorphic to G. This represents the rst superrigidity result pertaining to Group von Neumann algebras.
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Orbit inequivalent actions for Groups containing a copy of $\mathbb{F}_{2}$
Inventiones mathematicae, 2011Co-Authors: Adrian IoanaAbstract:We prove that if a Countable Group Γ contains a copy of $\mathbb{F}_{2}$ , then it admits uncountably many non orbit equivalent actions.
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Orbit inequivalent actions for Groups containing a copy of \mathbb{F}_{2}
Inventiones Mathematicae, 2010Co-Authors: Adrian IoanaAbstract:We prove that if a Countable Group Γ contains a copy of \(\mathbb{F}_{2}\), then it admits uncountably many non orbit equivalent actions.
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Ergodic Subequivalence Relations Induced by a Bernoulli Action
Geometric and Functional Analysis, 2010Co-Authors: Ionut Chifan, Adrian IoanaAbstract:Let Γ be a Countable Group and denote by $${\mathcal{S}}$$ the equivalence relation induced by the Bernoulli action $${\Gamma\curvearrowright [0, 1]^{\Gamma}}$$ , where [0, 1]^Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation $${\mathcal{R}}$$ of $${\mathcal{S}}$$ , there exists a partition { X _ i }_ i ≥0 of [0, 1]^Γ into $${\mathcal{R}}$$ -invariant measurable sets such that $${\mathcal{R}_{\vert X_{0}}}$$ is hyperfinite and $${\mathcal{R}_{\vert X_{i}}}$$ is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.
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Orbit inequivalent actions for Groups containing a copy of $\Bbb F_2$
arXiv: Group Theory, 2006Co-Authors: Adrian IoanaAbstract:We prove that if a Countable Group $\Gamma$ contains a copy of $\Bbb F_2$, then it admits uncountably many non orbit equivalent actions.
Sorin Popa - One of the best experts on this subject based on the ideXlab platform.
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A CLASS OF SUPERRIGID Group VON NEUMANN ALGEBRAS
Annals of Mathematics, 2013Co-Authors: Adrian Ioana, Sorin Popa, Stefaan VaesAbstract:We prove that for any Group G in a fairly large class of generalized wreath product Groups, the associated von Neumann algebra LG completely \remembers" the Group G. More precisely, if LG is isomorphic to the von Neumann algebra L of an arbitrary Countable Group , then must be isomorphic to G. This represents the rst superrigidity result pertaining to Group von Neumann algebras.
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on the fundamental Group of ii 1 factors and equivalence relations arising from Group actions
Quanta of Maths Clay Mathematics Proceedings, 2010Co-Authors: Sorin Popa, Stefaan VaesAbstract:Given a Countable Group G, we consider the sets Sfactor(G), Seqrel(G), of subGroups F R+ for which there exists a free ergodic probability measure preserving action G y X such that the fundamental Group of the associated II1 factor L 1 (X) oG, respectively orbit equivalence relation R(G y X), equals F. We prove that if G = 1 Z, with 6 1, then Sfactor(G) and Seqrel(G) contain R+ itself, all of its Countable subGroups, as well as unCountable subGroups that can have any Hausdor dimension 2 (0; 1). We deduce that there exist II1 factors of the form M = L 1 (X)oF1 such that the fundamental Group of M is R+, but M B(‘ 2 (N)) admits no continuous trace scaling action of R+. We then prove that if G = , with ; nitely generated ICC Groups, one of which has property (T), then Sfactor(G) = Seqrel(G) =ff1gg:
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on the fundamental Group of ii_1 factors and equivalence relations arising from Group actions
arXiv: Operator Algebras, 2008Co-Authors: Sorin Popa, Stefaan VaesAbstract:Given a Countable Group G, we consider the sets S_factor(G), S_eqrel(G), of subGroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental Group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial Group \Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its Countable subGroups, as well as unCountable subGroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\Gamma*\Lambda, with \Gamma, \Lambda finitely generated ICC Groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental Group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.
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Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid Groups
Inventiones mathematicae, 2007Co-Authors: Sorin PopaAbstract:We prove that if a Countable discrete Group Γ is w-rigid , i.e. it contains an infinite normal subGroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ= H × H ’ with H an infinite Kazhdan Group and H’ arbitrary), and $\mathcal{V}$ is a closed subGroup of the Group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ Countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space ( X ,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a Group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subGroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a Countable Group Λ is implemented by a conjugacy of the actions, with respect to some Group isomorphism Γ≃Λ.
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on the superrigidity of malleable actions with spectral gap
arXiv: Group Theory, 2006Co-Authors: Sorin PopaAbstract:We prove that if a Countable Group $\Gamma$ contains infinite commuting subGroups $H, H'\subset \Gamma$ with $H$ non-amenable and $H'$ ``weakly normal'' in $\Gamma$, then any measure preserving $\Gamma$-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli $\Gamma$-action) is cocycle superrigid. If in addition $H'$ can be taken non-virtually abelian and $\Gamma \curvearrowright X$ is an arbitrary free ergodic action while $\Lambda \curvearrowright Y=\Bbb T^\Lambda$ is a Bernoulli action of an arbitrary infinite conjugacy class Group, then any isomorphism of the associated II$_1$ factors $L^\infty X \rtimes \Gamma \simeq L^\infty Y \rtimes \Lambda$ comes from a conjugacy of the actions.
Adrien Le Boudec - One of the best experts on this subject based on the ideXlab platform.
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$$C^*$$ C ∗
Inventiones mathematicae, 2017Co-Authors: Adrien Le BoudecAbstract:A Countable Group is $$C^*$$ C ∗ -simple if its reduced $$C^*$$ C ∗ -algebra is simple. It is well-known that $$C^*$$ C ∗ -simplicity implies that the amenable radical of the Group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every Countable Group embeds into a Countable Group with trivial amenable radical and that is not $$C^*$$ C ∗ -simple.
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C^*-simplicity and the amenable radical
Inventiones Mathematicae, 2016Co-Authors: Adrien Le BoudecAbstract:A Countable Group is \(C^*\)-simple if its reduced \(C^*\)-algebra is simple. It is well-known that \(C^*\)-simplicity implies that the amenable radical of the Group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every Countable Group embeds into a Countable Group with trivial amenable radical and that is not \(C^*\)-simple.
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C*-simplicity and the amenable radical
arXiv: Group Theory, 2015Co-Authors: Adrien Le BoudecAbstract:A Countable Group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the Group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every Countable Group embeds into a Countable Group with trivial amenable radical and that is not C*-simple.
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Discrete Groups that are not C*-simple
arXiv: Group Theory, 2015Co-Authors: Adrien Le BoudecAbstract:A Countable Group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the Group must be trivial. We show that the converse does not hold by constructing explicit Countable Groups without non-trivial amenable normal subGroups and that are not C*-simple.
Brandon Seward - One of the best experts on this subject based on the ideXlab platform.
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Krieger's finite generator theorem for actions of Countable Groups II
Journal of Modern Dynamics, 2019Co-Authors: Brandon SewardAbstract:We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of Countable Groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abert–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every Countable Group admits a free ergodic action of positive Rokhlin entropy, we prove that: (ⅰ) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ⅱ) Bernoulli shifts have completely positive Rokhlin entropy; and (ⅲ) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.
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Krieger’s finite generator theorem for actions of Countable Groups I
Inventiones Mathematicae, 2018Co-Authors: Brandon SewardAbstract:For an ergodic p.m.p. action $$G \curvearrowright (X, \mu )$$ of a Countable Group G, we define the Rokhlin entropy $$h^{\mathrm {Rok}}_G(X, \mu )$$ to be the infimum of the Shannon entropies of Countable generating partitions. It is known that for free ergodic actions of amenable Groups this notion coincides with classical Kolmogorov–Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger’s finite generator theorem holds for all countably infinite Groups. Specifically, if $$h^{\mathrm {Rok}}_G(X, \mu ) < \log (k)$$ then there exists a generating partition consisting of k sets.
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borel structurability on the 2 shift of a Countable Group
Annals of Pure and Applied Logic, 2016Co-Authors: Brandon Seward, Robin TuckerdrobAbstract:Abstract We show that for any infinite Countable Group G and for any free Borel action G ↷ X there exists an equivariant class-bijective Borel map from X to the free part Free ( 2 G ) of the 2-shift G ↷ 2 G . This implies that any Borel structurability which holds for the equivalence relation generated by G ↷ Free ( 2 G ) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free ( 2 G ) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2 G lands in the free part. As a corollary we obtain that for every ϵ > 0 , every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ϵ. This generalizes a result of Danilenko and Park.
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Krieger's Finite Generator Theorem for Ergodic Actions of Countable Groups.
arXiv: Dynamical Systems, 2015Co-Authors: Brandon SewardAbstract:We continue the study of Rokhlin entropy, an isomorphism invariant for ergodic probability-measure-preserving actions of general Countable Groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Ab\'{e}rt--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every Countable Group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.
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borel structurability on the 2 shift of a Countable Group
arXiv: Dynamical Systems, 2014Co-Authors: Brandon Seward, Robin TuckerdrobAbstract:We show that for any infinite Countable Group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright 2^G$. This implies that any Borel structurability which holds for the equivalence relation generated by $G \curvearrowright \mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming from free Borel actions of $G$. A related consequence is that the Borel chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic numbers of free actions of $G$. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic $G$-equivariant map to $2^G$ lands in the free part. As a corollary we obtain that for every $\epsilon > 0$, every free pmp action of $G$ has a free factor which admits a $2$-piece generating partition with Shannon entropy less than $\epsilon$. This generalizes a result of Danilenko and Park.
Yoshikata Kida - One of the best experts on this subject based on the ideXlab platform.
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Stable actions and central extensions
Mathematische Annalen, 2017Co-Authors: Yoshikata KidaAbstract:A probability-measure-preserving action of a Countable Group is called stable if its transformation-Groupoid absorbs the ergodic hyperfinite equivalence relation of type \({\text {II}}_1\) under direct product. We show that for a Countable Group G and its central subGroup C, if G / C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.
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Stable actions of central extensions and relative property (T)
Israel Journal of Mathematics, 2015Co-Authors: Yoshikata KidaAbstract:Let us say that a discrete Countable Group is stable if it has an ergodic, free, probability-measure-preserving and stable action. Let G be a discrete Countable Group with a central subGroup C. We present a sufficient condition and a necessary condition for G to be stable. We show that if the pair (G, C) does not have property (T), then G is stable. We also show that if the pair (G, C) has property (T) and G is stable, then the quotient Group G/C is stable.
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Rigidity of amalgamated free products in measure equivalence
Journal of Topology, 2011Co-Authors: Yoshikata KidaAbstract:A discrete Countable Group \Gamma is said to be ME rigid if any discrete Countable Group that is measure equivalent to \Gamma is virtually isomorphic to \Gamma. In this paper, we construct ME rigid Groups by amalgamating two Groups satisfying rigidity in a sense of measure equivalence. A class of amalgamated free products is introduced, and discrete Countable Groups which are measure equivalent to a Group in that class are investigated.
