Amenable actions and groups

Amenable actions are answering the question: "When can we prevent things like the Banach-Tarski Paradox happening?". It turns out that the most intuitive measure-theoretic sufficient condition is also necessary. We will briefly discuss the paradox, prove the equivalent conditions for amenability, give some ways of producing interesting examples of amenable groups and talk about amenable groups which can't be produced in these 'elementary' ways.

Teaser question: show that you can't decompose Z into finitely many pieces, which after rearrangement by translations make two copies of Z. (I.e. that you can't get the Banach-Tarski paradox on Z.)