First $\ell^2$-Betti numbers and proper proximality
Operator Algebras
2022-11-14 v1 Group Theory
Abstract
We show that for a countable exact group, having positive first -Betti number implies proper proximality in this sense of \cite{BoIoPe21}. This is achieved by showing a cocycle superrigidty result for Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups are OE-superrigid.
Cite
@article{arxiv.2211.05951,
title = {First $\ell^2$-Betti numbers and proper proximality},
author = {Changying Ding},
journal= {arXiv preprint arXiv:2211.05951},
year = {2022}
}