Finite conjugacy classes and split exact cochain complexes

TL;DR

This paper investigates the relationship between finite conjugacy classes and the split exactness of cochain complexes in the context of isometric group actions on Banach spaces, revealing new rigidity phenomena.

Contribution

It establishes conditions under which cochain complexes are split exact based on the absence of almost invariant vectors, extending prior rigidity results.

Findings

Split exactness of cochain complexes under certain conditions

Absence of almost invariant vectors implies rigidity

Connections to prior work on group actions and cohomology

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 - Monod, Bader - Rosendal - Sauer and Nowak.