Cocycle Superrigidity and Group Actions on Stably Finite C*-Algebras

Abstract

Let $\Lambda $ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $\Lambda $ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $\Lambda $ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action via its (weak) 1-cohomology set endowed with a canonical pairing function. The key ingredient in this computation is Popa's cocycle superrigidity theorem for Bernoulli shifts on the hyperfinite II$_{1} $ factor $R$. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of $\Lambda $ on $D$ are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when $\Lambda $ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace. Finally, we use the techniques of this paper to construct outer actions on $R$ with prescribed cohomology. Precisely, for every infinite property (T) group $\Lambda$, and for every countable abelian group $\Gamma$, we construct an outer action of $\Lambda$ on $R$ whose 1-cohomology is isomorphic to $\Gamma$.