First ℓ2-Betti numbers and proper proximality

Abstract

We show that for a countable exact group, having positive first ℓ2-Betti number implies proper proximality in this sense of [3]. This is achieved by showing a cocycle superrigidity 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.