Some rigidity results for non-commutative Bernoulli shifts

We introduce the outer conjugacy invariants , for cocycle actions σ of discrete groups G on type II₁ factors N, as the set of real numbers t>0 for which the amplification σ^t of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly and the fundamental group of σ, , in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and σ is an action of G on the hyperfinite II₁ factor by Connes–Størmer Bernoulli shifts of weights {t_i}_i. Thus, and coincide with the multiplicative subgroup S of generated by the ratios {t_i/t_j}_i,j, while if S={1} (i.e. when all weights are equal), and otherwise. In fact, we calculate all the “1-cohomology picture” of σ^t,t>0, and classify the actions (σ,G) in terms of their weights {t_i}_i. In particular, we show that any 1-cocycle for (σ,G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on to the algebra pNp, for p a projection in N, p≠0,1, cannot be perturbed to a genuine action.