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,μ) such that has an invariant mean and that there is a Borel cocycle α: G × X → 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 Γ and Δ are measure equivalent discrete groups in the sense of Gromov, then such cocycles exist and Γ and Δ share the same weak amenability properties above.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-1-5,
author = {Paul Jolissaint},
title = {Proper cocycles and weak forms of amenability},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {73-87},
zbl = {1316.22003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-1-5}
}
Paul Jolissaint. Proper cocycles and weak forms of amenability. Colloquium Mathematicae, Tome 139 (2015) pp. 73-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm138-1-5/