  We study the cohomology of isometric group actions on (super) reflexive
Banach spaces with a focus on the relation between finite conjugacy classes and
split exactness of cochain complexes. In particular, we show that, if a
uniformly convex Banach module has no almost invariant vectors under the
FC-centre of the acting group, then the associated cochain complex is split
exact. Other similar rigidity results are established that are related to prior
work of Bader - Furman - Gelander - Monod, Bader - Rosendal - Sauer and Nowak.
