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Orbit full groups for locally compact groups

Authors: A. Carderi and F. Le Maître
Journal: Trans. Amer. Math. Soc. 370 (2018), 2321-2349
MSC (2010): Primary 37A15, 37A20; Secondary 22D10, 46L10
Published electronically: November 30, 2017
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Abstract: We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group.

We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

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Additional Information

A. Carderi
Affiliation: Institut für Geometrie, Technische Universität Dresden, 01062 Dresden, Germany

F. Le Maître
Affiliation: Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris cedex 13, France

Received by editor(s): January 31, 2016
Received by editor(s) in revised form: May 24, 2016
Published electronically: November 30, 2017
Additional Notes: The authors were partially supported by Projet ANR-14-CE25-0004 GAMME
The first author was partially supported by ERC Consolidator Grant No. 681207
Article copyright: © Copyright 2017 American Mathematical Society

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