Comptes Rendus
no. 10
Logic/Dynamical systems
On the classification problem of free ergodic actions of nonamenable groups

Presented by: the Editorial Board

Eusebio Gardella1; Martino Lupini2
1 Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
2 Mathematics Department, California Institute of Technology, 1200 E. California Blvd, 91125 Pasadena, CA, United States
Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1037-1040.

We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.

Nous montrons que, pour tout groupe dénombrable discret et non moyennable Γ, les relations de conjugaison, d'équivalence orbitale, d'équivalence orbitale stable, d'équivalence de von Neumann et d'équivalence de von Neumann stable des actions libres ergodiques de Γ sur un espace borélien standard muni d'une mesure de probabilité sans atomes ne sont pas Borel. Cela répond à une question de Kechris.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.10.004

Eusebio Gardella 1; Martino Lupini 2

1 Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
2 Mathematics Department, California Institute of Technology, 1200 E. California Blvd, 91125 Pasadena, CA, United States 
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Eusebio Gardella; Martino Lupini. On the classification problem of free ergodic actions of nonamenable groups. Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1037-1040. doi : 10.1016/j.crma.2017.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.004/

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