Countable abelian group actions and hyperfinite equivalence relations

Gao, Su; Jackson, Steve

doi:10.1007/s00222-015-0603-y

Cited by 45 publications

(113 citation statements)

References 10 publications

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“…This conjecture was first mentioned in [8] as a passing remark, and has gained traction since the settlement of the countable case [6]. Our Theorem 6.1 provides more evidence, beyond which the conjecture is wide open.…”

mentioning

confidence: 52%

“…Since each H m is a countable discrete abelian group, each E Zm H m is hyperfinite by the theorem of Gao and Jackson [6] …”

Section: Essentially Countable Orbit Equivalence Relationsmentioning

confidence: 99%

“…Both proofs use a theorem of Gao and Jackson [6] on the hyperfiniteness of orbit equivalence relations of countable abelian group actions. Theorem 1.3 can be viewed as a generalization of the Gao-Jackson theorem, since countable discrete groups are locally compact.…”

Section: 3 If G Is a Locally Compact Non-archimedean Abelian Polishmentioning

confidence: 99%

“…We say that an equivalence relation E is essentially hyperfinite if E ≤ B E 0 . It is a theorem of Gao and Jackson [6] that any Γ-orbit equivalence relation, where Γ is a countable discrete abelian group, is hyperfinite. We will use this theorem several times in the proofs below.…”

Section: 3mentioning

confidence: 99%

“…By a theorem of Gao and Jackson [6], the orbit equivalence relation of any countable abelian group is hyperfinite. …”

Section: Orbit Equivalence Relations Of Tame Groupsmentioning

confidence: 99%

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Non-archimedean abelian Polish groups and their actions

Ding

Gao

2017

Advances in Mathematics

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Abstract. In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group is tame if and only if it does not involve Z ω or (Z(p) <ω ) ω for any prime p. In addition to determining the structure of tame groups, we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the Borel reducibility hierarchy. It is shown that if such an orbit equivalence relation is essentially countable, then it must be essentially hyperfinite. We also find an upper bound in the Borel reducibility hierarchy for the orbit equivalence relations of all tame non-archimedean abelian Polish groups.

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mentioning

confidence: 52%

“…Since each H m is a countable discrete abelian group, each E Zm H m is hyperfinite by the theorem of Gao and Jackson [6] …”

Section: Essentially Countable Orbit Equivalence Relationsmentioning

confidence: 99%

Section: 3 If G Is a Locally Compact Non-archimedean Abelian Polishmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…By a theorem of Gao and Jackson [6], the orbit equivalence relation of any countable abelian group is hyperfinite. …”

Section: Orbit Equivalence Relations Of Tame Groupsmentioning