  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 through its (weak) 1-cohomology set endowed with
a canonical pairing function.
  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
$\Gamma$.
