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 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 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.

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.

Measurable chromatic numbers

2008 ◽  
Vol 73 (4) ◽  
pp. 1139-1157 ◽  
Author(s):  
Benjamin D. Miller
Keyword(s):  
Chromatic Number ◽  
Equivalence Relation ◽  
Initial Segment ◽  
Polish Space ◽  
Borel Function ◽  
Chromatic Numbers ◽  
Unilateral Shift ◽  
Dichotomy Theorem ◽  
Borel Functions ◽  
Measurable Chromatic Number

AbstractWe show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ]ℕ, although its Borel chromatic number is ℵ0. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for “the” minimum analytic graph with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ Є6 (2, 3…..ℵ0, c), there is a treeing of E0 with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm–Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (ℝ<ℕ, ⊇).

Borel complexity of isomorphism between quotient Boolean algebras

2008 ◽  
Vol 73 (4) ◽  
pp. 1328-1340
Author(s):  
Su Gao ◽  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Boolean Algebras ◽  
Equivalence Relations ◽  
Borel Set ◽  
Borel Sets ◽  
Opposite Case ◽  
Borel Complexity ◽  
Borel Ideal ◽  
The One ◽  
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.

scholarly journals EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE

2017 ◽  
Vol 82 (3) ◽  
pp. 893-930 ◽  
Author(s):  
WILLIAM CHAN
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Proper Forcing ◽  
Image Position

AbstractThe following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence relation.

Classifying Borel automorphisms

2007 ◽  
Vol 72 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
John D. Clemens
Keyword(s):  
Polish Space ◽  
Inverse Image ◽  
Product Topology ◽  
Cantor Space ◽  
Space Forms ◽  
Borel Set ◽  
Polish Spaces ◽  
Countable Space ◽  
Fixed Space ◽  
Borel Automorphism

§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

scholarly journals Forcing Properties of Ideals of Closed Sets

2011 ◽  
Vol 76 (3) ◽  
pp. 1075-1095 ◽  
Author(s):  
Marcin Sabok ◽  
Jindřich Zapletal
Keyword(s):  
Polish Space ◽  
Borel Function ◽  
Constant Property ◽  
Borel Set ◽  
Borel Sets ◽  
Combinatorial Properties ◽  
Closed Sets ◽  
Cohen Real ◽  
The Ideal

AbstractWith every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ -ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal.We also study the 1–1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1–1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

Ideals without ccc

1998 ◽  
Vol 63 (1) ◽  
pp. 128-148 ◽  
Author(s):  
Marek Balcerzak ◽  
Andrzej RosŁanowski ◽  
Saharon Shelah
Keyword(s):  
Polish Space ◽  
Borel Function ◽  
Borel Set ◽  
Borel Sets ◽  
Polish Groups ◽  
Perfect Set ◽  
Consistency Results ◽  
The Family ◽  
Whole Space ◽  
Condition B

AbstractLet I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X with f−1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family {B+x : x ∈ P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).

scholarly journals Trees and amenable equivalence relations

1990 ◽  
Vol 10 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Scot Adams
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Measure Space ◽  
Polynomial Growth ◽  
Finite Measure ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Borel Equivalence Relation

AbstractLet R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

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