The Ping-Pong Lemma, which historically was used to study free products of groups, is adapted to the setting of the homeomorphism group
of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criterion for embedding subgroups of $\mathrm{Homeo}_+(I)$ into Richard Thompson’s group $F$ . In particular, every member of our class of generating sets generates a group which embeds into $F$ and hence is not a free product. An analogous abstract theory is also developed for groups of permutations of an infinite set. This is joint work with Collin Bleak, Matt Brin, Martin Kassabov, and Matt Zaremsky.