Amenable versus hyperfinite borel equivalence relations

Journal of Symbolic Logic 58 (3):894-907 (1993)
  Copy   BIBTEX

Abstract

LetXbe a standard Borel space, and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK]. It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every probability measure μ onX, there is a Borel invariant setY⊆Xwith μ = 1 such thatE↾Y is hyperfinite. Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG. We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒFƒ. We have the following theorem.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,616

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

On Σ1 1 equivalence relations with Borel classes of bounded rank.Ramez L. Sami - 1984 - Journal of Symbolic Logic 49 (4):1273 - 1283.
Property τ and countable borel equivalence relations.Simon Thomas - 2007 - Journal of Mathematical Logic 7 (1):1-34.
Infinite Time Decidable Equivalence Relation Theory.Samuel Coskey & Joel David Hamkins - 2011 - Notre Dame Journal of Formal Logic 52 (2):203-228.
Amenable equivalence relations and Turing degrees.Alexander S. Kechris - 1991 - Journal of Symbolic Logic 56 (1):182-194.
New dichotomies for borel equivalence relations.Greg Hjorth & Alexander S. Kechris - 1997 - Bulletin of Symbolic Logic 3 (3):329-346.
Actions of non-compact and non-locally compact polish groups.Sławomir Solecki - 2000 - Journal of Symbolic Logic 65 (4):1881-1894.
Borel equivalence relations which are highly unfree.Greg Hjorth - 2008 - Journal of Symbolic Logic 73 (4):1271-1277.

Analytics

Added to PP
2009-01-28

Downloads
49 (#287,772)

6 months
5 (#247,092)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Countable borel equivalence relations.S. Jackson, A. S. Kechris & A. Louveau - 2002 - Journal of Mathematical Logic 2 (01):1-80.

Add more citations

References found in this work

Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.

Add more references