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