Given any group $G$, is there an amenable group $A(G)$ together with a morphism $G\rightarrow A(G)$, such that every other morphism $G\rightarrow A'$ to another amenable group $G'$ uniquely factorizes through $A'$?

That is the question. My approach would be to consider the set of normal subgroups with amenable quotient $S:=\{N\unlhd G|G/N $ is amenable $\}$. Then $\bigcap S$ is a normal subgroup of $G$. But I don't know, whether $G/\bigcap S$ is amenable. It embeds into the group $\prod_{H\in S}G/H$. It is not clear, that a infinite product of amenable groups is amenable again. But maybe one can embed $G/\bigcap S$ in a smaller group.

residually amenablequotient of $G$. – Andreas Thom Oct 4 '10 at 16:38