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

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##### Date

2017-01-17##### MFO Scientific Program

OWLF 2016##### Series

Oberwolfach Preprints;2017,01##### Author

Gardella, Eusebio

Lupini, Martino

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Show full item record##### OWP-2017-01

##### 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
$\Gamma$.

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