An equivalence relation is a relation which looks like ordinary equality of numbers, but which may hold between other kinds of objects. This extends previous results obtained by gaojackson for abelian groups and by jacksonkechris. Locally nilpotent groups and hyperfinite equivalence relations. 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. Is the borel reduction of an hyperfinite equivalence. The classification of finite borel equivalence relations on the. 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. The infinite tensor product of a countable number of factors of type i n with respect to.
The intersection of any two different cells is empty. So, up to a set of measure 0, e is the ion of an ascending sequence of finite equivalence relations. 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. Is the borel reduction of an hyperfinite equivalence relation. 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. Declare two animals related if they can breed to produce fertile o spring. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. Pdf countable abelian group actions and hyperfinite equivalence. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. Amenable and hyperfinite equivalence relations any action of a countable group on a standard borel space gives rise to a countable borel equivalence relation and, conversely, any countable borel equivalence relation can be generated as the orbit equivalence relation of some group action. In particular, the outer automorphism group of any countable group is hyperfinite. 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.
Then r is an equivalence relation and the equivalence classes of r are the. Equivalence relations you can have a relation which simultaneously has more than one of the properties we have been discussing. 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. Given a borel action of a countable group on a polish space x, we denote by ex the orbit equivalence relation of y x, the borel equivalence relation. And again, equivalence sub f immediately inherits the properties of equality, which makes it an equivalence relation. An equivalence relation on a set s, is one that satisfies the following three properties for all x, y, z math\inmath s. It is of course enormously important, but is not a very interesting example, since no two distinct objects are related by equality.
Ordinal definability and combinatorics of equivalence. The classification of hypersmooth borel equivalence relations. Why the hell does standard borel space redirect here. In section 4 we discuss certain aspects of the geometry of abelian groups and. 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. 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.
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. Measure reducibility of countable borel equivalence relations. Let e be an aperiodic, nonsmooth hyperfinite borel equivalence relation. These properties are true for equivalence classes with respect to any equivalence relation. An amenable equivalence relation is generated by a single. Given an action of gon x, the ex gequivalence class of xis called the orbit of xand is equal to gx g x.
The structure of hyperfinite borel equivalence relations. Suppose that r is a hyperfinite equivalence relation on x, b. Define a relation on s by x r y iff there is a set in f which contains both x and y. X, there exists a nonsingular transformation t of x such that, up to a null set. The proof is found in your book, but i reproduce it here. 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. Equivalence relation, in mathematics, a generalization of the idea of equality between elements of a set. That is, any two equivalence classes of an equivalence relation are either mutually disjoint or identical.
Amenable versus hyperfinite borel equivalence relations. 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. Equivalence relations now we group properties of relations together to define new types of important relations. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. On sofic actions and equivalence relations sciencedirect.
These were introduced in this context in kechris 91 by adapting. A wider class than the hyperfinite equivalence relations consists of the so called amenable ones. Equivalence relations mathematical and statistical sciences. Intuitively, a treeing of r is a measurablyvarying way of makin each equivalence class into the vertices of a tree. Equivalence relation mathematics and logic britannica.
Instead of a generic name like r, we use symbols like. On constructing ergodic hyperfinite equivalence relations of. Trees and amenable equivalence relations ergodic theory. What is the equivalence class of this equivalence relation. On constructing ergodic hyperfinite equivalence relations. 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. We show that for any polish group g and any countable normal subgroup. Then the equivalence classes of r form a partition of a. 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. The shannonmcmillanbreiman theorem beyond amenable groups. It follows that any two cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.
Quotients by countable subgroups are hyperfinite joshua frisch and forte shinko abstract. The group e preserves the measure a d is ergodic with respect to a. 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. 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. Equivalence relations r a is an equivalence iff r is. 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.
Trees and amenable equivalence relations ergodic theory and. 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. The classification of hyperfinite borel equivalence relations. Given an equivalence class a, a representative for a is an element of a, in. 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. If xy and yz then xz this holds intuitively for when. Hyperfinite equivalence relations and the union problem. They also observe as part of this analysis that every hyperfinite equivalence relation is treeable and every smooth countable borel equivalence. Let rbe an equivalence relation on a nonempty set a. Given an equivalence class a, a representative for a is an element of a, in other words it is a b2xsuch that b. A countable borel equivalence relation e on a standard borel space x is hyperfinite if there is an increasing sequence f0.
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 hyperfinite type ii 1 factor also arises from the groupmeasure space construction for ergodic free measurepreserving actions of countable amenable groups on probability spaces. In this article we establish the following theorem. Let x be a standard borel space and e a borel equivalence relation on x. How would you apply the idea to a whole relationset. Countable borel equivalence relations semantic scholar. Hyperfinite borel equivalence relations 195 3 is the notion of hyperfiniteness effective, i. We call e hyperfinite if there is a borel automorphism t of x such that xey. Request pdf on constructing ergodic hyperfinite equivalence relations of nonproduct type product type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit. Equivalence relations are a way to break up a set x into a union of disjoint subsets. A relation r on a set x is an equivalence relation if it is i re. 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. If a is a set, r is an equivalence relation on a, and a and b are elements of a, then either a \b.
In particular, the outer automorphism group of any countable group is hyper nite. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Knowing of a computation in one group, the isomorphism allows us to perform the analagous computation in the other group. Countable abelian group actions and hyperfinite equivalence. More interesting is the fact that the converse of this statement is true. Here are three familiar properties of equality of real numbers. Our proof explicitly constructs topological generators for the orbit equivalence relation of the. Pdf countable abelian group actions and hyperfinite.
The indecomposable characters on a group gare in onetoone correspondence with the. There is an extensive literature on the subject of countable borel equivalence. Mat 300 mathematical structures equivalence classes and. 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. Let rbe an equivalence relation on a nonempty set a, and let a. Abstractlet x be the space of all infinite 0,1sequences and e be the tail equivalence relation on x. Foliations of polynomial growth are hyperfinite springerlink. In general if eis any equivalence relation on x, we write xe for the eequivalence class of x.