Ergodic Subequivalence Relations Induced by a Bernoulli Action

By I. Chifan, Adrian Ioana. 2008

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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 {Xi}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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b57f03dd75abda61555714df6905fa5bf3668815

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In Geometry and Topology

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Citations in the corpus

Bass-Serre rigidity results in von Neumann algebras (2008, background): “menable andvv∗(A′ 0∩ M)vv∗ is amenable as well. Since Ahas no amenable direct summand, and A⊂ A′ 0∩M, it follows that A′ 0 ∩Mhas no amenable direct summand either. We get a contradiction. We refer to [6] for some applications of the notion of solid action in ergodic theory. 6.3. Applications to Orbi”
W*-superrigidity for Bernoulli actions of property (T) groups (2010, unknown): “The proof of [8], Lemma 5 provides a family {Ai}i∈I of von Neumann subalgebras of M1 and a family {Γi}i∈I of finite subgroups of Γ1 such that Ai ⊂ A Γi, for all i ∈ I, and L(2)(M̃1) L(2)(M1) ∼= ⊕ i∈I L (2)(〈M1, eAi〉), as Hilbert M1–bimodules.”
Orbit Equivalence and Measured Group Theory (2010, background): “However, Chifan-Ioana [CI10] extending an argument of Abert-Nikolov [AN07] proved that this is indeed the case when the commutant of Γ ( X, µ ) in Aut( X, µ ) acts ergodically on ( X, µ ).”
RIGIDITY FOR VON NEUMANN ALGEBRAS (2017, background): “igidity theorems for proﬁnite actions [Io08,Fu09,GITD16], leading to an explicit uncountable family of Borel incomparable treeable equivalence relations [Io13]. •solid ergodicity of Bernoulli actions [CI08] (see also [Oz04]). 1.2. The classiﬁcation of II1 factors is governed by a strong amenable/non”
On cocycle superrigidity for Gaussian actions (2009, methodology): “The deformation described below was first used by Ioana [19] in the case when the base space is non-amenable, and later used by Chifan and Ioana [6] in part to obtain solidity of L(X, μ)oσ 0, whenever L0 is solid and 0 y (X, μ) is the Bernoulli shift.”