The Structure of Hyperfinite Borel Equivalence Relations

Abstract

Abstract. We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed. This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the hyperfinite ones. These are by definition the increasing unions of sequences of Borel equivalence relations with finite equivalence classes but equivalently they can be also described as the ones induced by the orbits of a single Borel automorphism.

Linear algebraic groups and countable Borel equivalence relations

“…There is thus a great variety of interesting examples of (up to ∼ B ) countable Borel equivalence relations occurring in many areas of mathematics. See, for example, the papers [19], [22], [23], [24], [5], [18] and the references contained therein, for further information concerning the descriptive study of countable Borel equivalence relations. The facts we are about to state in the next paragraph come from the last paper.…”

“…By Serre [31], Theorem 6 in page 36 and Definitions 7, 4 and Proposition 3 in pp. 32-33, the form 5 , g 6 , x 7 ) over Q p for some form g. Then if the matrix M ∈ GL 7 (Q p ) implements the equivalence and SO F is the Q-group of matrices in GL 7 preserving…”

Conjugacy equivalence relation on subgroups

“…The reader is referred to [5] for basic facts on descriptive set theory, while more extensive introductions to the theory of Borel equivalence relations can be found in [1] or [4].…”

On the complexity of the quasi-isometry and virtual isomorphism problems for finitely generated groups

“…We shall begin by describing the precise Borel complexity of the isomorphism relation ∼ = on G. Recall that an equivalence relation E on a Polish space X is said to be countable iff every E-class is countable. By DoughertyJackson-Kechris [4], there exists a universal countable Borel equivalence relation E ∞ ; i.e. a countable Borel equivalence relation E ∞ such that F ≤ B E ∞ for every countable Borel equivalence relation F .…”