On the Kropholler conjecture

This short note, produced for the volume of conjectures produced on the occasion of Guido Mislin's retirement, describes the status of the Kropholle conjecture which may be viewed as a far reaching generalisation of Stallings' theorem on groups with more than one end

Approximation properties for discrete groups

We give an overview of approximation properties in the context of operator algebras associated with groups

Property A and exactness of the uniform Roe algebra

In this short note, prepared for the volume of conjectures to celebrate Guido Mislin's retirement, we outline the conjecture that a uniformly discrete bounded geometry metric space X has property A if and only if the uniform Roe algebra C^?(X ) is exact

A homological characterization of topological amenability

Generalizing Block and Weinberger’s characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of G on its Stone-Cech compactification we obtain a homological characterisation of exactness of the group