The Ping-pong lemma for infinite cyclic subgroups

Statement

Let G be a group acting on a set X. Let a₁, ...,a_k be elements of G of infinite order, where k ≥ 2. Suppose there exist disjoint nonempty subsets

of $X$ with the following properties:

• $a_{i}(X - X_{i}^{-}) \subseteq X_{i}^{+}$ for $i = 1, ..., k$;
• $a_{i}^{-1}(X - X_{i}^{+}) \subseteq X_{i}^{-}$ for $i = 1, ..., k$.

Then the subgroup $H = \langle a_{1}, ..., a_{k}\rangle \le G$ generated by a₁, ..., a_k is free with free basis $\{a_{1}, ..., a_{k}\}$.