scholarly journals Amenable versus hyperfinite Borel equivalence relations

1993 ◽  
Vol 58 (3) ◽  
pp. 894-907 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Countable Group ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

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.

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.

scholarly journals Inner amenability, property Gamma, McDuff factors and stable equivalence relations

2017 ◽  
Vol 38 (7) ◽  
pp. 2618-2624 ◽  
Author(s):  
TOBE DEPREZ ◽  
STEFAAN VAES
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Countable Group ◽  
Acta Math ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Von Neumann ◽  
Stable Equivalence ◽  
Inner Amenability

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

scholarly journals Incomparable actions of free groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2084-2098
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER
Keyword(s):  
Probability Measure ◽  
Free Group ◽  
Equivalence Relation ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Free Groups ◽  
Borel Equivalence Relation ◽  
Separability Condition ◽  
Borel Probability ◽  
Free Actions

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

Borel equivalence relations which are highly unfree

2008 ◽  
Vol 73 (4) ◽  
pp. 1271-1277 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Probability Space ◽  
Ergodic Measure ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Borel Probability

AbstractThere is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E∣c is not essentially free on any conull C ⊂ X.

scholarly journals Pointwise ergodic theorems beyond amenable groups

2012 ◽  
Vol 33 (3) ◽  
pp. 777-820 ◽  
Author(s):  
LEWIS BOWEN ◽  
AMOS NEVO
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Countable Group ◽  
Equivalence Relations ◽  
Ergodic Theorems ◽  
General Group ◽  
Amenable Groups ◽  
Stable Type ◽  
Amenable Action ◽  
Weakly Mixing

AbstractWe prove pointwise and maximal ergodic theorems for probability-measure-preserving (PMP) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable typeIII$_1$. We show that this class contains all irreducible lattices in connected semi-simple Lie groups without compact factors. We also establish similar results when the stable type isIII$_\lambda $,$0 \lt \lambda \lt 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of PMP actions of amenable groups to include PMP amenable equivalence relations. Secondly, we show that it is possible to reduce the proof of ergodic theorems for PMP actions of a general group to the proof of ergodic theorems in an associated PMP amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

scholarly journals Cofinal families of Borel equivalence relations and quasiorders

2005 ◽  
Vol 70 (4) ◽  
pp. 1325-1340 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Compact Metric Spaces ◽  
Borel Equivalence Relation ◽  
Borel Ideal ◽  
Borel Reducibility

AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

scholarly journals BROOKS’ THEOREM FOR MEASURABLE COLORINGS

2016 ◽  
Vol 4 ◽  
Author(s):  
CLINTON T. CONLEY ◽  
ANDREW S. MARKS ◽  
ROBIN D. TUCKER-DROB
Keyword(s):  
Probability Measure ◽  
Cayley Graphs ◽  
Borel Probability Measure ◽  
Group Actions ◽  
New Technique ◽  
Borel Space ◽  
Standard Borel Space ◽  
List Colorings ◽  
A New Technique ◽  
Borel Probability

We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$-cliques, then $G$ admits a ${\it\mu}$-measurable $d$-coloring with respect to any Borel probability measure ${\it\mu}$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.

scholarly journals Countable sections for locally compact group actions

1992 ◽  
Vol 12 (2) ◽  
pp. 283-295 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Compact Group ◽  
Equivalence Relation ◽  
Group Actions ◽  
Locally Compact Group ◽  
Borel Space ◽  
Locally Compact ◽  
Borel Equivalence Relation ◽  
Measurable Section ◽  
Singular Action

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

scholarly journals Stability of products of equivalence relations

2018 ◽  
Vol 154 (9) ◽  
pp. 2005-2019 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Probability Measure ◽  
Direct Product ◽  
Equivalence Relation ◽  
Equivalence Relations ◽  
Similar Question ◽  
Stable Equivalence ◽  
Local Characterization

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

Sign in / Sign up

Export Citation Format

Share Document