Structurable equivalence relations

Chen, Ruiyuan; Kechris, Alexander S.

doi:10.4064/fm428-7-2017

Cited by 8 publications

(6 citation statements)

References 28 publications

(76 reference statements)

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“…These are all open problems. 3 We draw attention to a pair of questions which we find particularly interesting. [28,Theorem 3.7].…”

Section: Results In Reverse Mathematicsmentioning

confidence: 99%

Uniformity, universality, and computability theory

Marks

2017

J. Math. Log.

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We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups.We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal.We also show the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

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“…These are all open problems. 3 We draw attention to a pair of questions which we find particularly interesting. [28,Theorem 3.7].…”

Section: Results In Reverse Mathematicsmentioning

confidence: 99%

Uniformity, universality, and computability theory

Marks

2017

J. Math. Log.

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“…Then, by Proposition 5.2.4, E × I N is induced by a Borel action of Γ. On the other hand, E × I N cannot be induced by a free Borel action of Γ, since if that was the case then E × I N ⊑ i B E, contradicting the Addendum following [CK18,5.28].…”

Section: Consider Now the Equivalence Relationmentioning

confidence: 96%

Realizations of countable Borel equivalence relations

Frisch¹,

Kechris²,

Shinko³

et al. 2021

Preprint

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We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as K σ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.

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“…There is a canonical connection between subequivalence relations of the equivalence relation E a induced by an action a ∈ A(Γ, X, µ) and IRE on Γ, which is a special case of structurability of such equivalence relations. See [KM,29.1], [CK,Section 2], and [T-D, Appendix A] for the particular case of equivalence relations.…”

Section: A Selection Theorem For Hyperfinitenessmentioning

confidence: 99%