Finite conjugacy classes and split exact cochain complexes

Abstract

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 and Monod (2007), Bader, Rosendal and Sauer (2014) and Nowak (2017).