Bernoulli actions are weakly contained in any free action

Abért, Miklós; Weiss, Benjamin

doi:10.1017/s0143385711000988

Cited by 43 publications

(69 citation statements)

References 9 publications

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“…A similar rephrasing also applies to weak containment of measure preserving actions, as pointed out by Abért-Weiss [AW11]. If we view a finite Borel partition A 0 , .…”

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confidence: 79%

“…Several invariants of measure preserving actions such as groupoid cost [AW11] ( [Ke10] for the case of free actions) and independence number [CK10] are known to increase or decrease with weak containment (see also [AE11a] and [CKT-D11] for other examples). A consequence of Theorem 1.5 is that, for a finitely generated group Γ, among all nonatomic measure preserving actions of type θ, the groupoid cost attains its maximum and the independence number attains its minimum on s θ,λ .…”

Section: Is the Bernoulli Shift Action Of γ Then ι × S γ Is A Free Amentioning

confidence: 99%

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Weak equivalence and non-classifiability of measure preserving actions

Tucker-Drob

2013

Ergod. Th. Dynam. Sys.

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Abstract. Abért-Weiss have shown that the Bernoulli shift s Γ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that s Γ × a is weakly equivalent to a. Using random Bernoulli shifts introduced by Abért-Glasner-Virag we generalized this to non-free actions, replacing s Γ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth's theory of turbulence to show that the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence on the weak equivalence class of a free measure preserving action do not admit classification by countable structures. This in particular shows that there are no free weakly rigid actions, i.e., actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek.We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group Γ is said to have property EMD * if the action p Γ of Γ on its profinite completion weakly contains all ergodic measure preserving actions of Γ, and Γ is said to have property MD if ι × p Γ weakly contains all measure preserving actions of Γ, where ι denotes the identity action on a standard non-atomic probability space. Kechris shows that EMD * implies MD and asks if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

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“…A similar rephrasing also applies to weak containment of measure preserving actions, as pointed out by Abért-Weiss [AW11]. If we view a finite Borel partition A 0 , .…”

mentioning

confidence: 79%

Section: Is the Bernoulli Shift Action Of γ Then ι × S γ Is A Free Amentioning

confidence: 99%

Weak equivalence and non-classifiability of measure preserving actions

Tucker-Drob

2013

Ergod. Th. Dynam. Sys.

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“…Thus, we have used the first element in our stack [0,1] N to set up an Γ-equivariant colouring of the set of cycles. We now will use the other elements in the stack to set up an infinite recursive Γ-equivariant cutting procedure which will have the property, that it does not disconnect the graph G. Let's pick a standard enumeration α : N × N ∼ = N and set ψ x := α • ϕ x .…”

Section: Relative Minimal Spanning Forestsmentioning

confidence: 99%

“…Note that the set ψ x (0), we cut the edge with the maximal label (this is the second in our stack [0,1] N ) not in L(a). Clearly, this does not disconnect G.…”

Section: Relative Minimal Spanning Forestsmentioning

confidence: 99%

The expected degree of minimal spanning forests

Thom

2016

Combinatorica

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Abstract. We give a lower bound on the expected degree of the free minimal spanning forest of a vertex transitive graph in terms of its spectral radius. This result answers a question of Lyons-Peres-Schramm and simplifies the Gaboriau-Lyons proof of the measurable-group-theoretic solution to von Neumann's problem.In the second part we study a relative version of the free minimal spanning forest. As a consequence of this study we can show that non-torsion unitarizable groups have fixed price one.

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“…In [AW13], M. Abért and B. Weiss exhibit a remarkable anti-rigidity phenomenon for weak containment by showing that every free p.m.p. action of Γ weakly contains the Bernoulli action over an atomless base space.…”

Section: Introductionmentioning

confidence: 99%

Weak containment rigidity for distal actions

Ioana

Tucker-Drob

2016

Advances in Mathematics

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Abstract. We prove that if a measure distal action α of a countable group Γ is weakly contained in a strongly ergodic probability measure preserving action β of Γ, then α is a factor of β. In particular, this applies when α is a compact action.As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.

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Bernoulli actions are weakly contained in any free action

Cited by 43 publications

References 9 publications

Weak equivalence and non-classifiability of measure preserving actions

Weak equivalence and non-classifiability of measure preserving actions

The expected degree of minimal spanning forests

Weak containment rigidity for distal actions

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