Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

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.