First $\ell^2$-Betti numbers and proper proximality

TL;DR

This paper establishes a link between positive first 2-Betti numbers and proper proximality in countable exact groups, and demonstrates superrigidity results for Bernoulli shifts of certain non-properly proximal groups.

Contribution

It proves that positive first 2-Betti numbers imply proper proximality and shows cocycle superrigidity for Bernoulli shifts of non-properly proximal groups.

Findings

Positive first 2-Betti number implies proper proximality.

Bernoulli shifts of certain groups are OE-superrigid.

Cocycle superrigidity for Bernoulli shifts of non-properly proximal groups.

Abstract

We show that for a countable exact group, having positive first $\ell^2$-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.