Rigidity theorems for actions of product groups and countable Borel equivalence relations

Hjorth, Greg; Kechris, Alexander S.

doi:10.1090/memo/0833

Cited by 59 publications

(75 citation statements)

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“…It is easy to see that the existence of almost invariant sets implies the existence of almost invariant vectors for the Koopman representation κ 0 (look at the characteristic functions), but the converse may fail, as was first proved by Schmidt [11] (for another example, see Hjorth-Kechris [7,Theorem A3.2]). In fact, the existence of almost invariant sets depends only on the orbit equivalence relation, which, in the ergodic case, is equivalent to non-E 0 -ergodicity (Jones-Schmidt [9]), while the existence of almost invariant vectors depends on the group action (see [7] again).…”

Section: µ)mentioning

confidence: 95%

Amenable actions and almost invariant sets

Kechris

Tsankov

2007

Proc. Amer. Math. Soc.

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Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on M X , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ M X has almost invariant sets.

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Section: µ)mentioning

confidence: 95%

Amenable actions and almost invariant sets

Kechris

Tsankov

2007

Proc. Amer. Math. Soc.

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“…, Λ l be non-trivial finitely generated torsion-free groups with k ≤ l. Suppose that all the Γ i 's are non-elementary hyperbolic groups and that the direct products Γ 1 × · · · × Γ k and Λ 1 × · · · × Λ l are measure equivalent. Then k = l. More strongly, after permutation of the indices, Γ i is measure equivalent to Λ i for all i. Hjorth-Kechris [44] showed a similar result. Theorem 1.2 can be regarded as an analogue of these results.…”

Section: Introductionmentioning

confidence: 76%

The mapping class group from the viewpoint of measure equivalence theory

Kida

2008

Memoirs of the AMS

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We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.

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“…Condition 2.1(ii) is reminiscent of the conclusion of the "unique ergodicity argument" first introduced by Adams [1] in the measure-theoretical setting and later exploited by Thomas [16,17] and Hjorth-Kechris [7]. Of course, the following result is an immediate consequence of Theorem 2.1 and implies that ≡ T is not countable universal.…”

Section: Introductionmentioning

confidence: 89%