Description:
Let ʌ be a countably infinite property (T) group, and let A be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of ʌ onA. The proof consists in assigning, to any second countable abelian pro-p group G, a strongly outer action of ʌ on A 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 II1 factor. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of ʌ on A are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when ʌ is only assumed to contain an infinite subgroup with relative property (T), and A is a (not necessarily simple) separable, nuclear,UHF-absorbing, self-absorbing C*-algebra with at least one trace.