 Ping-pong lemma

Suppose nontrivial groups $G_1,G_2$ act on a set and there are disjoint nonempty subsets $X_1,X_2$ such that every nonidentity element of $G_1$ maps $X_2$ into $X_1$, while every nonidentity element of $G_2$ maps $X_1$ into $X_2$. Assume at least one factor has at least three elements. Then the subgroup generated by $G_1$ and $G_2$ is their free product. The cardinality hypothesis excludes the counterexample in which $G_1=G_2=C_2$ acts by swapping $X_1$ and $X_2$. Versions allowing both factors to have order two need an additional separation hypothesis.

The Ping-Pong Lemma is a result in geometric group theory that is often used to prove that a group is free, or to show that a group has a particular property, such as being non-abelian or having a certain type of subgroup. The lemma is particularly useful in the context of groups acting on trees or hyperbolic spaces.