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Conjugacy, orbit equivalence and classification of measure-preserving group actions

Published online by Cambridge University Press:  01 June 2009

ASGER TÖRNQUIST
ASGER TÖRNQUIST*
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Room 6092, Toronto, Ontario, Canada (email: [email protected])
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Abstract

We prove that if G is a countable discrete group with property (T) over an infinite subgroup H≤G which contains an infinite Abelian subgroup or is normal, then G has continuum-many orbit-inequivalent measure-preserving almost-everywhere-free ergodic actions on a standard Borel probability space. Further, we obtain that the measure-preserving almost-everywhere-free ergodic actions of such a G cannot be classified up to orbit equivalence by a reasonable assignment of countable structures as complete invariants. We also obtain a strengthening and a new proof of a non-classification result of Foreman and Weiss for conjugacy of measure-preserving ergodic almost-everywhere-free actions of discrete countable groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 1033 - 1049
Copyright
Copyright © Cambridge University Press 2008

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