Proper cocycles and weak forms of amenability

Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,μ)$ such that $L^\infty(X,μ)$ has an invariant mean and that there is a Borel cocycle $α:G\times X\rightarrow H$ which is proper in a suitable, natural sense. 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 $G$. We observe that it is the case for a weak form of measure equivalence for pairs of discrete groups.

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