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