Some applications of superrigidity to Borel
                    equivalence relations

Thomas, Simon

doi:10.1090/dimacs/058/10

Cited by 9 publications

(5 citation statements)

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“…However, no countable Borel equivalence relations beyond E 0 were known even to be indivisible into two pieces until we showed this for E ∞ in [30]. In contrast, we note that Thomas [39] has shown the existence of a countable Borel equivalence relation E on an uncountable standard Borel space that is not 2-indivisible. Indeed, there is a countable Borel equivalence relation E where E < B E ⊕E, where E ⊕ E is the direct sum of two disjoint copies of E.…”

Section: Theorem 325 Given Any Uniformly Universal Countable Borel Eq...mentioning

confidence: 74%

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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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Section: Theorem 325 Given Any Uniformly Universal Countable Borel Eq...mentioning

confidence: 74%

“…For instance, this is a typical proof strategy in applications of cocycle superrigidity to the field of countable Borel equivalence relations. See for example [1,16,39,41,46].…”

Section: Introductionmentioning

confidence: 99%

Uniformity, universality, and computability theory

Marks

2017

J. Math. Log.

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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%

Martin’s conjecture and strong ergodicity

Thomas

2009

Arch. Math. Logic

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“…In fact [Th,Lemma 3.4] gives a countable Borel equivalence relation E ∈ N SC, which is not divisible by any n > 1, i.e., there is no F ∈ C with nF ∼ B F . It follows, using Corollary 2.5, that there is a fam on [N SC], + for which ϕ([E]) = 1, so ϕ takes a finite value.…”

Section: Cardinal Algebrasmentioning

confidence: 99%