Classification and Structure of Cocycles of Amenable Ergodic Equivalence Relations

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

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On full groups of non-ergodic probability-measure-preserving equivalence relations

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.20 ◽

2015 ◽

Vol 36 (7) ◽

pp. 2218-2245 ◽

Cited By ~ 2

Author(s):

FRANÇOIS LE MAÎTRE

Keyword(s):

Probability Measure ◽

Operator Algebras ◽

Baire Category ◽

Equivalence Relations ◽

Continuity Property ◽

Algebraic Characterization ◽

Full Group ◽

Automatic Continuity ◽

Ergodic Equivalence Relations ◽

The Cost

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

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Quasinormal subrelations of ergodic equivalence relations

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-98-04909-0 ◽

1998 ◽

Vol 126 (11) ◽

pp. 3361-3370 ◽

Cited By ~ 3

Author(s):

Alexandre I. Danilenko

Keyword(s):

Equivalence Relations ◽

Ergodic Equivalence Relations

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Subrelations of ergodic equivalence relations

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004958 ◽

1989 ◽

Vol 9 (2) ◽

pp. 239-269 ◽

Cited By ~ 23

Author(s):

J. Feldman ◽

C. E. Sutherland ◽

R. J. Zimmer

Keyword(s):

Finite Index ◽

Lie Group ◽

Equivalence Relation ◽

Equivalence Relations ◽

Type Ii ◽

Compact Lie Group ◽

Rigidity Results ◽

Higher Rank ◽

Ergodic Equivalence Relations ◽

Inner Automorphisms

AbstractWe introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) ofQon it; in the case whereQis amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

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