# The structure of hy-per nite Borel equivalence relations

@inproceedings{Dougherty1994TheSO, title={The structure of hy-per nite Borel equivalence relations}, author={Randall L. Dougherty and Steve Jackson and Alexander S. Kechris}, year={1994} }

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…

## 229 Citations

### The classification of hypersmooth Borel equivalence relations

- Mathematics
- 1997

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…

### Borel homomorphisms of smooth -ideals

- Mathematics
- 2007

Given a countable Borel equivalence relation E on a Polish space, letIE denote the -ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism fromIE toIF if…

### Amenable versus hyperfinite Borel equivalence relations

- MathematicsJournal of Symbolic Logic
- 1993

By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X so that every equivalence class [X]_E is countable.

### Linear algebraic groups and countable Borel equivalence relations

- Mathematics
- 2000

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…

### Cardinal characteristics and countable Borel equivalence relations

- MathematicsMath. Log. Q.
- 2017

A family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$.

### Borel asymptotic dimension and hyperfinite equivalence relations

- Mathematics
- 2020

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. In…

### Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations

- Mathematics
- 2005

This Memoir is both a contribution to the theory of Borel equivalence relations,
considered up to Borel reducibility, and measure preserving group actions
considered up to orbit equivalence. Here E…

### Countable Borel equivalence relations

- Mathematics
- 2020

These notes cover an introduction to countable Borel equivalence relations. The theory of countable Borel equivalence relation is a huge subject. It has a broad impact in several areas of mathematics…

### Realizations of countable Borel equivalence relations

- Mathematics
- 2019

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples…

### Property τ and Countable Borel Equivalence Relations

- MathematicsJ. Math. Log.
- 2007

For each prime q, it is proved that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

## References

SHOWING 1-10 OF 42 REFERENCES

### An equivalence relation that is not freely generated

- Mathematics
- 1988

In this paper it is shown that there exists a Borel equivalence relation with countable equivalence classes that is not generated by a free Borel action of a countable discrete group. In Feldman and…

### On non-singular transformations of a measure space. I

- Mathematics
- 1969

We consider a Lebesgue measure space (M, ∇, m). By an automorphism of (M, ∇, m) we mean a bi-measurable transformation of (M, ∇, m) that together with its inverse is non-singular with respeot to m.…

### On quasi-invariant measures in uniquely ergodic systems

- Mathematics
- 1971

Let X be a compact metrizable space, ~ (X) its G-algebra of Borel sets, and let T be a homeomorphism of X. We denote for a Borel measure # on X by T/2 the measure that is obtained by setting…

### On orbit equivalence of borel automorphisms

- Mathematics
- 1989

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with…

### Countable sections for locally compact group actions

- MathematicsErgodic Theory and Dynamical Systems
- 1992

Abstract It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete…

### A Glimm-Effros dichotomy for Borel equivalence relations

- Mathematics
- 1990

A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in…

### The Structure of Borel Equivalence Relations in Polish Spaces

- Mathematics
- 1992

An exposition of recent work on Borel equivalence relations in Polish spaces is presented. This includes a general Glimm-Effros dichotomy for Borel equivalence relations and a study of countable…

### Amenable equivalence relations and Turing degrees

- PhilosophyJournal of Symbolic Logic
- 1991

In [12] Slaman and Steel posed the following problem: Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order…

### Essays in Commutative Harmonic Analysis

- Mathematics
- 1979

This book considers various spaces and algebras made up of functions, measures, and other objects-situated always on one or another locally compact abelian group, and studied in the light of the…

### An amenable equivalence relation is generated by a single transformation

- MathematicsErgodic Theory and Dynamical Systems
- 1981

Abstract We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set: It follows that any two…