Bernoulli actions of type III1 and L2-cohomology

Vaes, Stefaan; Wahl, Jonas

doi:10.1007/s00039-018-0438-y

Cited by 35 publications

(70 citation statements)

References 33 publications

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“…Bernoulli actions were obtained in [VW17], confirming the conjecture for 'most' countable groups. Although a general dissipative-conservative criterion was established in [VW17], it remained an open problem to prove ergodicity and determine the type in full generality.…”

Section: Several Families Of Examples Of Type IIIsupporting

confidence: 65%

“…Only very recently, in [VW17], the first results were established for nonamenable groups G. It was conjectured in [VW17] that a countable group G admits a Bernoulli action of type III if and only if the first L 2 -cohomology H 1 (G, 2 (G)) is nonzero, which is equivalent to saying that G is either infinite amenable or has positive first L 2 -Betti number. The connection with L 2 -cohomology stems from the following observation: if the marginal measures µ g satisfy µ g (0) ∈ [δ, 1 − δ] for all g ∈ G and some δ > 0, then by Kakutani's criterion (see [Kak48]), the corresponding Bernoulli action is nonsingular if and only c g (h) = µ h (0) − µ g −1 h (0) has the property that c g ∈ 2 (G) for all g ∈ G, thus defining a 1-cocycle c ∈ Z 1 (G, 2 (G)).…”

Section: Introductionmentioning

confidence: 99%

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Ergodicity and type of nonsingular Bernoulli actions

2020

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We determine the Krieger type of nonsingular Bernoulli actions G g∈G ({0, 1}, µ g ). When G is abelian, we do this for arbitrary marginal measures µ g . We prove in particular that the action is never of type II ∞ if G is abelian and not locally finite, answering Krengel's question for G = Z. When G is locally finite, we prove that type II ∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II 1 or III 1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group G admits a Bernoulli action of type III 1 if and only if G has nontrivial first L 2 -cohomology.

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Section: Several Families Of Examples Of Type IIIsupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Ergodicity and type of nonsingular Bernoulli actions

2020

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“…Added in the proof. Since this paper was posted on the arXiv in April 2017, Problem 3 above has been solved in [VW17]. Indeed, it is shown in [VW17, Section 7] that for a large class of nonsingular Bernoulli actions of the free groups F n with n ≥ 3, the induced orbit equivalence relation is strongly ergodic and has prescribed Sd and τ invariants.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Strongly ergodic equivalence relations: spectral gap and type III invariants

Houdayer¹,

Marrakchi²,

Verraedt³

2017

Ergod. Th. Dynam. Sys.

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We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable 1-cocycles with values into locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and τ invariants for type III strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type III1 ergodic equivalence relation R, the Maharam extension c(R) is strongly ergodic if and only if R is strongly ergodic and the invariant τ (R) is the usual topology on R. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes' structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and τ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

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“…We give a positive (partial) answer to this question in Theorem 3 by producing type-III λ Bernoulli action of G for λ ∈ (0, 1). The case where λ = 1 is known from [6,57].…”

Section: Countable Amenable Groupsmentioning

confidence: 99%

“…Both of their constructions are Bernoulli shifts on two symbols, where the corresponding probabilities are defined inductively. See also Vaes and Wahl [57,Section 6] for examples of type-III 1 nonsingular Bernoulli shifts where the probabilities are specified by an explicit formula. Type-III 1 Markov shifts also play a crucial role in Kosloff's construction of type-III 1 Anosov diffeomorphisms [39,40].…”

Section: Introductionmentioning

confidence: 99%