Equivalence relations you can have a relation which simultaneously has more than one of the properties we have been discussing. Why the hell does standard borel space redirect here. Measure reducibility of countable borel equivalence relations. We show that for any polish group g and any countable normal subgroup. Given an equivalence class a, a representative for a is an element of a, in. By definition, the full group of the equivalence relation e is the group e all borel automorphisms s of x, such that xesx for all x ax. Countable abelian group actions and hyperfinite equivalence. Our proof explicitly constructs topological generators for the orbit equivalence relation of the. Trees and amenable equivalence relations ergodic theory and. Request pdf on constructing ergodic hyperfinite equivalence relations of nonproduct type product type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit.
Then the equivalence classes of r form a partition of a. Abstractlet x be the space of all infinite 0,1sequences and e be the tail equivalence relation on x. A wider class than the hyperfinite equivalence relations consists of the so called amenable ones. Equivalence relation, in mathematics, a generalization of the idea of equality between elements of a set. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. It follows that any two cartan subalgebras of a hyperfinite factor are conjugate by an automorphism. An amenable equivalence relation is generated by a single. Ordinal definability and combinatorics of equivalence. Countable borel equivalence relations semantic scholar.
Equivalence relations r a is an equivalence iff r is. Hyperfinite borel equivalence relations 195 3 is the notion of hyperfiniteness effective, i. Get pdf 497 kb abstract a long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by. If is an equivalence relation on x, and px is a property of elements of x, such that whenever x y, px is true if py is true, then the property p is said to be welldefined or a class invariant under the relation a frequent particular case occurs when f is a function from x to another set y. The shannonmcmillanbreiman theorem beyond amenable groups. And again, equivalence sub f immediately inherits the properties of equality, which makes it an equivalence relation. On sofic actions and equivalence relations sciencedirect. The intersection of any two different cells is empty. A countable borel equivalence relation e on a standard borel space x is hyperfinite if there is an increasing sequence f0. Foliations of polynomial growth are hyperfinite springerlink. It is of course enormously important, but is not a very interesting example, since no two distinct objects are related by equality. My research interests lie in descriptive set theory and its connections to related areas such as computability theory, combinatorics, ergodic theory, probability, and operator algebras. Mat 300 mathematical structures equivalence classes and.
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Our main results in this paper provide a classification of hyperfinite borel equivalence relations under two different notions of equivalence. Get pdf 497 kb abstract a long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a borel action of a countable amenable group is hyperfinite. Instead of a generic name like r, we use symbols like. In section 3 we prove a borel marker lemma similar to gj, lemma 2. A relation r on a set x is an equivalence relation if it is i re.
Hyperfinite equivalence relations and the union problem. Here are three familiar properties of equality of real numbers. The group e preserves the measure a d is ergodic with respect to a. Given an action of gon x, the ex gequivalence class of xis called the orbit of xand is equal to gx g x. Let x be a standard borel space and e a borel equivalence relation on x. Let r be a borel equivalence relation with countable equivalence classes on a measure space m. That is, any two equivalence classes of an equivalence relation are either mutually disjoint or identical. An equivalence relation is a relation which looks like ordinary equality of numbers, but which may hold between other kinds of objects. The classification of finite borel equivalence relations on the. These properties are true for equivalence classes with respect to any equivalence relation.
We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. Two elements of the set are considered equivalent with respect to the equivalence relation if and only if they are elements of the same cell. The hyperfinite type ii 1 factor also arises from the groupmeasure space construction for ergodic free measurepreserving actions of countable amenable groups on probability spaces. The structure of hyperfinite borel equivalence relations. Let rbe an equivalence relation on a nonempty set a. An equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations en where all enequivalence classs are finite. Equivalence relation mathematics and logic britannica. Then r is an equivalence relation and the equivalence classes of r are the. An equivalence relation on a set s, is one that satisfies the following three properties for all x, y, z math\inmath s.
How would you apply the idea to a whole relationset. And the theorem that we have is that every relation r on a set a is an equivalence relation if and only if it in fact is equal to equivalence sub f for some function f. There is an extensive literature on the subject of countable borel equivalence. Is the borel reduction of an hyperfinite equivalence relation. We show that for any polish group g and any countable normal subgroup g, the coset equivalence relation g is a hyper nite borel equivalence relation. The classification of hyperfinite borel equivalence relations. A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a borel action of a countable amenable group is hyperfinite. They also observe as part of this analysis that every hyperfinite equivalence relation is treeable and every smooth countable borel equivalence. The infinite tensor product of a countable number of factors of type i n with respect to their tracial states is the hyperfinite type ii 1 factor. Intuitively, a treeing of r is a measurablyvarying way of makin each equivalence class into the vertices of a tree. The proof is found in your book, but i reproduce it here. We also extend a result of dye for countable groups to show that if a locally compact second countable group g acts freely on a lebesgue space x with finite invariant measure, so that. In this article we establish the following theorem.
Pdf countable abelian group actions and hyperfinite. Pdf an equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations en. Characters on the full group of an ergodic hyperfinite. In section 4 we discuss certain aspects of the geometry of abelian groups and. Equivalence relations and functions october 15, 20 week 14 1 equivalence relation a relation on a set x is a subset of the cartesian product x. Is the borel reduction of an hyperfinite equivalence. On constructing ergodic hyperfinite equivalence relations. Locally nilpotent groups and hyperfinite equivalence relations. The indecomposable characters on a group gare in onetoone correspondence with the. An equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations en where all en equivalence classs are finite. The classification of hypersmooth borel equivalence relations. Amenable versus hyperfinite borel equivalence relations. This paper develops the foundations of the descriptive set theory of countable borel equivalence relations on polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.
More interesting is the fact that the converse of this statement is true. These were introduced in this context in kechris 91 by adapting. If a is a set, r is an equivalence relation on a, and a and b are elements of a, then either a \b. Standard borel spaces and kuratowkis theorem have small connection with borel equivalence relation and may defintely not be considered as a subtopic of it. The classification of hyperfinite borel equivalence. Equivalence relations now we group properties of relations together to define new types of important relations. This extends previous results obtained by gaojackson for abelian groups and by jacksonkechris. The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. We prove that for any amenable nonsingular countable equivalence relation r. X, there exists a nonsingular transformation t of x such that, up to a null set.
In particular, the outer automorphism group of any countable group is hyperfinite. Declare two animals related if they can breed to produce fertile o spring. A countable borel equivalence relation is called hyperfinite if it is induced by a borel zaction, i. We give an elementary proof that there are two topological generators for the full group of every aperiodic hyper nite probability measure preserving borel equivalence relation. Equivalence relations are a way to break up a set x into a union of disjoint subsets. Let e be an aperiodic, nonsmooth hyperfinite borel equivalence relation. So, up to a set of measure 0, e is the ion of an ascending sequence of finite equivalence relations. If xy and yz then xz this holds intuitively for when. The infinite tensor product of a countable number of factors of type i n with respect to. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. We call e hyperfinite if there is a borel automorphism t of x such that xey. In particular, the outer automorphism group of any countable group is hyper nite.
In general if eis any equivalence relation on x, we write xe for the eequivalence class of x. The notion of an amenable equivalence relation was introduced by zimmer 23. Knowing of a computation in one group, the isomorphism allows us to perform the analagous computation in the other group. Thus, when two groups are isomorphic, they are in some sense equal. What is the equivalence class of this equivalence relation. Given an equivalence class a, a representative for a is an element of a, in other words it is a b2xsuch that b. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. Quotients by countable subgroups are hyperfinite joshua frisch and forte shinko abstract. Suppose that r is a hyperfinite equivalence relation on x, b.
Locally nilpotent groups and hyperfinite equivalence relations 3 nilpotent groups. On constructing ergodic hyperfinite equivalence relations of. Trees and amenable equivalence relations ergodic theory. Pdf countable abelian group actions and hyperfinite equivalence. Observe that in our example the equivalence classes of any two elements are either the same or are disjoint have empty intersection and, moreover, the union of all equivalence classes is the entire set x.
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