Amenable actions and almost invariant sets

Kechris, Alexander S.; Tsankov, Todor

doi:10.1090/s0002-9939-07-09116-2

Cited by 34 publications

(47 citation statements)

References 11 publications

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“…(This fact was observed earlier in [KT08] Proposition 2.4). If f C / ∈ I g then gf C = f C, i.e., f −1 gf ∈ C which is equivalent to g ∈ f Cf −1 .…”

supporting

confidence: 72%

Every countably infinite group is almost Ornstein

Bowen

2012

Dynamical Systems and Group Actions

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“…(This fact was observed earlier in [KT08] Proposition 2.4). If f C / ∈ I g then gf C = f C, i.e., f −1 gf ∈ C which is equivalent to g ∈ f Cf −1 .…”

supporting

confidence: 72%

Every countably infinite group is almost Ornstein

Bowen

2012

Dynamical Systems and Group Actions

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“…However, for arbitrary X 0 , this is not the case. It is shown in Kechris-Tsankov [18] that for generalized Bernoulli shifts, 1 Γ ≺ κ 0 implies the existence of almost invariant sets, while it is open as to whether the same holds for actions on compact Polish groups by automorphisms.…”

Section: Amenable Koopman Representations and Almost Invariant Vectorsmentioning

confidence: 99%

Modular actions and amenable representations

Epstein¹,

Tsankov

2009

Trans. Amer. Math. Soc.

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Abstract. Consider a measure-preserving action Γ (X, µ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y ) with countable image, where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones.

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“…The following theorem seems to have been known for a while (see [13] or [17] Theorem 2.1.) We include a simple proof for completeness.…”

Section: Lemma 22 If G Is An Arbitrary D-regular Graphing Thenmentioning

confidence: 99%