Ping-pong lemma

Statement in terms of two elements and free groups
Suppose $$G$$ is a group with a group action on a set $$X$$. Further, suppose $$A$$ and $$B$$ are two subsets of $$X$$ such that neither is contained in the other. Further, suppose $$a,b \in G$$ are elements such that:


 * $$b^n(A) \subseteq B \ \forall \ n \ne 0$$.
 * $$a^n(B) \subseteq A \ \forall \ n \ne 0$$.

Then, the subgroup $$\langle a, b \rangle$$ is a free group with $$\{a,b\}$$ as a freely generating set.

Statement in terms of two subgroups and free products
Suppose $$G$$ is a group with a group action on a set $$X$$. Further, suppose $$A$$ and $$B$$ are two subsets of $$X$$ such that neither is contained in the other. Further, suppose $$H, K \le G$$ are subgroups such that:


 * $$g(A) \subseteq B \ \forall \ g \in K$$.
 * $$g(B) \subseteq A \ \forall \ g \in H$$.

Then, the subgroup $$\langle H, K \rangle$$ is an internal free product of the subgroups $$H$$ and $$K$$.

Applications

 * Upper-triangular and lower-triangular unipotent matrices generate free non-abelian subgroup in special linear group over integers