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 . Then the subgroup generated by and is their free product.