We investigate free product structures in R. Thompson's group V , primarily by studying the topological dynamics associated with V 's action on the Cantor Set. We show that the class of free products which can be embedded into V includes the free product of any two finite groups, the free product of any finite group with Q/Z, and the countable non-abelian free groups. We also show the somewhat surprising result that Z 2 * Z does not embed in V , even though V has many embedded copies of Z 2 and has many embedded copies of free products of pairs of its subgroups.Question 1. Is it true that if G and H are non-trivial subgroups of V , with | G |≥ 3 and G, H ∼ = G * H in V , then there are distinct non-empty sets P G and P H in C so that for any non-trivial elements g ∈ G and h ∈ H we have P H g ⊂ P G and P G h ⊂ P H ?Colloquially, must every free product of groups in V arise from a Ping-Pong in V ? Let CF PV be the smallest class of groups which contains B and which is closed under 1. isomorphism, 2. passing to subgroups,