Ping-pong lemma for several subgroups

Statement

This version of the ping-pong lemma ensures that several subgroups of a group acting on a set generate a free product. The following statement appears in Olijnyk and Suchchansky (2004), and the proof is from de la Harpe (2000).

Let G be a group acting on a set X and let H₁, H₂, ..., H_k be subgroups of G where k ≥ 2, such that at least one of these subgroups has order greater than 2. Suppose there exist pairwise disjoint nonempty subsets $X_{1}, X_{2}, ...,X_{k}$ of $X$ such that the following holds:

• For any $i \ne s$ and for any $h$ in $H_{i}$, $h \ne 1$ we have $h(X_{s}) \subseteq X_{i}$.

Then

$$\langle H_1,\dots, H_k\rangle=H_1\ast\dots \ast H_k.$$