Amenability of a free product
= Amenability of a free product
For nontrivial groups $A,B$, the <free product> $A*B$ is amenable exactly in the exceptional case $A\cong B\cong C_2$. In that case it is the <infinite dihedral group>, hence a <virtually cyclic group> and amenable. In every other case the <normal form theorem for a free product> and the <ping-pong lemma> produce a subgroup isomorphic to $F_2$; the fact that <a subgroup of an amenable group is amenable> then obstructs amenability.