real analysis - Countable discrete abelian group amenable

The direct limit $G = bigcup_n G_n$ of a nested sequence of countable amenable groups $G_n$ is still amenable, since every finite set $S$ in $G$ will lie in one of the $G_n$ and thus there must exist some finite set $F_S$ which is not shifted very much by $S$. Since there are only a countable number of $S$, one can diagonalise and obtain a Folner sequence for $G$.

Since every countable abelian group is the direct limit of finitely generated abelian groups, which have polynomial growth and are thus amenable, every countable abelian group is amenable.

An instructive example is the free group $bigcup_n Z^n$ on countably many generators. Here, the sets ${-N_n,ldots,N_n}^n$ will form a Folner sequence if $N_n$ grows sufficiently rapidly in n.

I have some notes on amenability that cover these topics at

http://terrytao.wordpress.com/2009/04/14/some-notes-on-amenability/