Actions of non-compact and non-locally compact Polish groups

2000 ◽  
Vol 65 (4) ◽  
pp. 1881-1894 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Borel Equivalence Relations ◽  
Polish Groups ◽  
Local Compactness ◽  
Polish Group

AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

2006 ◽  
Vol 71 (4) ◽  
pp. 1108-1124 ◽  
Author(s):  
Alex Thompson
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Invariant Metrics ◽  
Orbit Equivalence ◽  
Invariant Metric ◽  
Continuous Action ◽  
Polish Groups ◽  
Polish Group ◽  
Left Invariant Metrics

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals Diagonal actions and Borel equivalence relations

2006 ◽  
Vol 71 (4) ◽  
pp. 1081-1096 ◽  
Author(s):  
Longyun Ding ◽  
Su Gao
Keyword(s):  
Equivalence Relation ◽  
Abelian Groups ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Borel Equivalence Relations ◽  
Polish Groups

AbstractWe investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.

A dichotomy theorem for turbulence

2002 ◽  
Vol 67 (4) ◽  
pp. 1520-1540 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Borel Function ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Borel Set ◽  
Polish Group ◽  
Strengthened Version ◽  
Dichotomy Theorem ◽  
Image Position

In this note we show:Theorem 1.1. Let G be a Polish group and X a Polish G-space with the induced orbit equivalence relation EG Borel as a subset of X × X. Then exactly one of the following:(I) There is a countable languageℒand a Borel functionsuch that for all x1, x2 ∈ Xor(II) there is a turbulent Polish G-space Y and a continuous G-embeddingThere are various bows and ribbons which can be woven into these statements. We can strengthen (I) by asking that θ also admit a Borel orbit inverse, that is to say some Borel functionfor some Borel set B ⊂ Mod(ℒ), such that for all x ∈ Xand then after having passed to this strengthened version of (I) we still obtain the exact same dichotomy theorem, and hence the conclusion that the two competing versions of (I) are equivalent. Similarly (II) can be relaxed to just asking that τ be a Borel G-embedding, or even simply a Borel reduction of the relevant orbit equivalence relations. It is in fact a consequence of 1.1 that all the plausible weakenings and strengthenings of (I) and (II) are respectively equivalent to one another.I will not closely examine these possible variations here. The equivalences alluded to above follow from our main theorem and the results of [3]. That monograph had previously shown that (I) and (II) are incompatible, and proved a barbaric forerunner of 1.1, and gone on to conjecture the dichotomy result above.

scholarly journals Analytic equivalence relations and Ulm-type classifications

1995 ◽  
Vol 60 (4) ◽  
pp. 1273-1300 ◽  
Author(s):  
Greg Hjorth ◽  
Alexander S. Kechris
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Borel Measure ◽  
Polish Space ◽  
Borel Function ◽  
Equivalence Relations ◽  
Cantor Space ◽  
Continuous Action ◽  
Borel Set ◽  
Continuous Embedding

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).Moreover, alternative (II) is equivalent to:(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

Actions by the classical Banach spaces

2000 ◽  
Vol 65 (1) ◽  
pp. 392-420 ◽  
Author(s):  
G. Hjorth
Keyword(s):  
Symmetric Group ◽  
Banach Spaces ◽  
Polish Space ◽  
Equivalence Relations ◽  
Continuous Group ◽  
Locally Compact ◽  
Polish Spaces ◽  
Polish Group ◽  
Quotient Spaces ◽  
Continuous Actions

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that theBorel cardinalitiesof the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.One of the first things to be addressed concerns the complexity of these equivalence relations. This question forappears in [1].

scholarly journals Decompositions and measures on countable Borel equivalence relations

2020 ◽  
pp. 1-33
Author(s):  
RUIYUAN CHEN
Keyword(s):  
Equivalence Relation ◽  
Countable Group ◽  
Equivalence Relations ◽  
Uniform Measure ◽  
Ergodic Decomposition ◽  
Continuous Action ◽  
Borel Sets ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Measure Preserving Actions

Abstract We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

2013 ◽  
Vol 56 (1) ◽  
pp. 136-147
Author(s):  
Radu-Bogdan Munteanu
Keyword(s):  
Equivalence Relation ◽  
Product Type ◽  
Bratteli Diagram ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Property A

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

scholarly journals Uniformity, universality, and computability theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750003
Author(s):  
Andrew S. Marks
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Technical Tool ◽  
Free Products ◽  
Borel Sets ◽  
Borel Equivalence Relations ◽  
Strengthened Form ◽  
Shift Action ◽  
Natural Classes ◽  
Main Technical Tool

We prove a number of results motivated by global questions of uniformity in computabi- lity 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.

Bi-Borel reducibility of essentially countable Borel equivalence relations

2005 ◽  
Vol 70 (3) ◽  
pp. 979-992 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space ◽  
Borel Reducibility ◽  
New Feature ◽  
Borel Probability ◽  
Image Position

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

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