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VOL. 73 | 2017 Splitting in orbit equivalence, treeable groups, and the Haagerup property
Yoshikata Kida

Editor(s) Koji Fujiwara, Sadayoshi Kojima, Ken'ichi Ohshika

## Abstract

Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.

## Information

Published: 1 January 2017
First available in Project Euclid: 4 October 2018

zbMATH: 07272050
MathSciNet: MR3728498

Digital Object Identifier: 10.2969/aspm/07310167