Nonamenability of a nonabelian free group
= Nonamenability of a nonabelian free group
The <free group> $F_2=\langle a,b\rangle$ is not an <amenable group>. Partitioning reduced words by their first letters gives two translated decompositions of $F_2$ whose invariance equations add to $2=1$, contradicting the existence of an <invariant mean>.