Amenability of abelian and solvable groups

Remarks

For part 1, the means on $\ell^\infty(G)$ form a nonempty weak-$*$ compact convex set, and each $g \in G$ acts on it by the continuous affine map $m \mapsto (f \mapsto m(g^{-1} \cdot f))$. These maps commute because $G$ is abelian, so the Markov–Kakutani fixed-point theorem gives a common fixed point, which is an invariant mean. For part 2, amenability passes to extensions: if $N$ is a normal subgroup with $N$ and $G/N$ amenable, then $G$ is amenable. A solvable group has a finite series $\set{1} = G_k \le \cdots \le G_0 = G$ with each $G_{i+1}$ normal in $G_i$ and $G_i/G_{i+1}$ abelian, so induction on $k$ gives part 2.

Example: the group of upper triangular $2 \times 2$ real matrices with determinant $1$ is solvable, hence amenable as a discrete group. In contrast, $\mathrm{SL}(2, \Z)$ is not amenable, since it contains a free group of rank $2$.

The same statements hold for locally compact groups: every abelian and every solvable locally compact group is amenable.

Going deeper

For a student

This result, together with the amenability of finite groups and stability under subgroups, quotients, extensions and directed unions, generates the class of elementary amenable groups. It contrasts with the non-amenability of free groups. By the Tits alternative, a finitely generated linear group is either virtually solvable, hence amenable, or contains a free group of rank $2$.