Proper cocycles and weak forms of amenability

Paul Jolissaint¹

¹ Institut de Mathématiques Université de Neuchâtel É.-Argand 11 CH-2000 Neuchâtel, Switzerland

Colloquium Mathematicum, Tome 138 (2015) no. 1, pp. 73-87

Résumé

Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard space $(X,\mu )$ such that $L^\infty (X,\mu )$ has an invariant mean and that there is a Borel cocycle $\alpha :G\times X\rightarrow H$ which is proper in the sense of Jolissaint (2000) and Knudby (2014). We show that if $H$ has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does $G$. In particular, we show that if $\varGamma $ and $\varDelta $ are measure equivalent discrete groups in the sense of Gromov, then such cocycles exist and $\varGamma $ and $\varDelta $ share the same weak amenability properties above.

Détail

DOI : 10.4064/cm138-1-5

Keywords: locally compact second countable groups assume acts measure class preserving standard space infty has invariant mean there borel cocycle alpha times rightarrow which proper sense jolissaint knudby has three properties haagerup property a t menability weak amenability weak haagerup property does particular vargamma vardelta measure equivalent discrete groups sense gromov cocycles exist vargamma vardelta share weak amenability properties above