a paradoxical decomposition of a group

Just my curiosity... Are there proofs the following fact, which does not involve Hall's matching theorem:

A group $\Gamma$ is amenable if and only if it does not admit a paradoxical decomposition.

Def: A group $\Gamma$ has a paradoxical decomposition if there are pairwise disjoint subsets $F_1,\ldots, F_n$, $E_1,\ldots, E_m$ of $\Gamma$, and elements $g_1,\ldots g_n$, $h_1,\ldots,h_m\in \Gamma$ such that $\Gamma$ can be expressed as $$\Gamma= \bigsqcup_{i} g_i F_i= \bigsqcup_{j} h_j E_j$$

According to a note in Grigorchuk's and Sunic's Self-Similarity and Branching in Group Theory, there is a proof not using the matching theorem in the book The Banch-Tarski Paradox by Stan Wagon. By the way, you have to mention that $\Gamma$ is also the union of $E_1,...,E_m,F_1,...,F_n$.

• Thank you, Sven, your reference is new for me. I think it is OK to skip the condition that the group is the union of these sets, since it is clear that the condition that I've wrote and existence of left-invariant mean contradict to each other. thus from this condition follows that the group is not amenable. the other way needs more work. Jun 14, 2011 at 12:11