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The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups

Gardella, Eusebio and Lupini, Martino (2017) The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups. . (Submitted)

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Building on work of Popa, Ioana, and Epstein--Törnquist, we show that, for every nonamenable countable discrete group Γ, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing) measure preserving actions of Γ on the standard atomless probability space are not Borel, thus answering questions of Kechris. This is an optimal and definitive result, which establishes a neat dichotomy with the amenable case, since any two free ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent by classical results of Dye and Ornstein--Weiss. The statement about conjugacy solves the nonamenable case of Halmos' conjugacy problem in Ergodic Theory, originally posed by Halmos in 1956 for ergodic transformations. In order to obtain these results, we study ergodic (or weak mixing) class-bijective extensions of a given ergodic countable probability measure preserving equivalence relation R. When R is nonamenable, we show that the relations of isomorphism and von Neumann equivalence of extensions of R are not Borel. When R is amenable, all the extensions of R are again amenable, and hence isomorphic by classical results of Dye and Connes--Feldman--Weiss. This approach allows us to extend the results about group actions mentioned above to the case of nonamenable locally compact unimodular groups, via the study of their cross-section equivalence relations.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Additional Information:The first-named author was partially funded by SFB 878 Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation. The second-named author was partially supported by the NSF Grant DMS- 1600186. This work was initiated while the authors were visiting the Centre de Recerca Matemàtica in March 2017, in occasion of the Intensive Research Programme on Operator Algebras. Part of the work was done while the authors where visiting the University of Houston in July and August 2017, in occasion of the Workshop on Applications of Model Theory to Operator Algebras, supported by the NSF grant DMS-1700316. The authors gratefully acknowledge the hospitality of both institutions.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
Subject Keywords:Orbit equivalence, von Neumann equivalence, pmp action, pmp expansion, nonamenable group, Bernoulli action, cocycle superrigidity, pmp groupoid
Classification Code:2000 Mathematics Subject Classification. Primary 37A20, 03E15; Secondary 20L05, 37A55
Record Number:CaltechAUTHORS:20180410-130304133
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85726
Deposited By: Tony Diaz
Deposited On:11 Apr 2018 14:55
Last Modified:03 Oct 2019 19:34

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