This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel automorphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism. An equivalence relation E on X is Borel reducible to an equivalence relation F on Y if there is a Borel map f: X → Y with xEy ⇔ f(x)Ff(y). We write then E ≤ F. If E ≤ Fand F ≤ E we say that E,F are Borel bi-reducible, in symbols E ≈ ∗ F.When E ≈ ∗ Fthe quotient spaces X/E, Y/F have the same “effective ” or “definable ” cardinality. We say that E,F are Borel isomorphic if there exists a Borel bijection f: X → Y with xEy ⇔ f(x)Ff(y). Below we denote by E0,Et the equivalence relations on the Cantor space 2 N given by: xE0y ⇔

\Lambda \Lambda

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory

Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators. The proofs are group-theoretic, and we refer to descriptive set theory only for the relevant definitions and for motivation for the results. Introduction Given a class K of structures for a fixed first order language L, one may ask what kinds of complete invariants can be used to classify the elements of K up to isomorphism. For those classes consisting of the countable models of some L ! 1 ;! -sentence, Friedman and Stanley [FS] proposed to use the methods of descriptive set theory to study their possible invariants and defined the notion of Borel reducibility between such classes of structures. In [HK], Hjorth and Kechris continued this study and situated it within the general the...

Introduction. These notes are based upon a day-long lecture workshop presented by Simon

Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank n are incomparable with respect to Borel reducibility.

Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.

The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book Becker-Kechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same.

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c: X → I such that xEy ⇔ c(x) =c(y). For this to be of any interest both I and c must be explicit or definable and as simple and concrete as possible. The theory of Borel equivalence relations studies the set-theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems. In organizing this study, the following concept of reducibility is fundamental. Let E,F be equivalence relations on standard Borel spaces X, Y, resp. We say that E is Borel reducible to F,insymbols, E ≤B F,

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.