This post is a place to list a number of papers that have recently appeared on the arXiv which reformulate the notion of exactness for groups (or property A for spaces or amenable actions) in different ways:

- A cohomological characterisation of Yu’s Property A for metric spaces by Brodzki, Niblo and Wright.
- Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak.
- Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright.
- A note on topological amenability by Monod.

I’ll post more later about the relations between these.