# Ping-pong lemma

## Statement

### 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 $K$.