Topics in orbit equivalence
type: Item
instance of: scholarly article
title: Topics in orbit equivalence
zbMATH Open document ID: 1058.37003
author: Benjamin D. Miller
author: Alexander S. Kechris
published in: Lecture Notes in Mathematics
publication date: 2004-12-14T00:00:00Z
review text: These notes provide a rapid but thorough description of orbit equivalence with both ergodic theory and the perspective of set theory in mind. Orbit equivalence is concerned with the equivalence relation induced on a measure space \(X\) by a measure-preserving or measure-class-preserving action of a group \(\Gamma\) (points are equivalent if one is in the orbit of the other). These notes deal with the classical setting, in which \(\Gamma\) is countable and discrete, and \(X\) is a standard Borel space. The authors introduce two useful thematic notions. The first is `elasticity', exemplified by the extension by \textit{D. S. Ornstein} and \textit{B. Weiss} [Bull. Am. Math. Soc., New Ser. 2, 161--164 (1980; Zbl 0427.28018)] of \textit{H. A. Dye}'s theorem [Am. J. Math. 81, 119--159 (1959; Zbl 0087.11501) and Am. J. Math. 85, 551--576 (1963; Zbl 0191.42803)]: any two Borel nonatomic probability preserving actions of amenable groups are orbit equivalent. Thus the equivalence relation induced by such an action does not remember anything about the group that created it apart from the vague recollection that it was amenable. The second is `rigidity', in which the equivalance relation induced by a group action carries a great deal of information about the group. For example, Furman has shown that if a free, nonatomic, probability-preserving action of some countable group \(\Gamma\) is orbit equivalent to the natural action of \(SL(3,\mathbb Z)\) on the \(3\)-torus, then \(\Gamma\) is isomorphic to \(SL(3,\mathbb Z)\) and the actions are Borel isomorphic. A more recent instance of `rigidity' is a result of Gaboriau, which gives the statement (under similar hypotheses) that orbit equivalence for actions of free groups remembers the rank of the free group. These notes are in three parts. The first gives a quick treatment of ergodic theory. The second deals with orbit equivalence in the presence of amenability -- the `elastic' world: Dye's theorem, the Ornstein-Weiss theorem on …
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 2122439
zbMATH Keywords: orbit equivalence
zbMATH Keywords: amenability
zbMATH Keywords: costs
MaRDI profile type: Publication
full work available at URL: b99421
OpenAlex ID: W1488189778
DOI: 10.1007/B99421
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description: scientific article; zbMATH DE number 2122439