Free products in R. Thompson’s group V

We investigate some product structures in R. Thompson's group V, primarily by studying the topological dynamics associated with V's action on the Cantor set C. We draw attention to the class D_(V,C) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D_(V,C) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G≤V and H ∈ D_(V,C), then G~H embeds into V. Finally, if G, H ∈ D_(V,C), then G*H embeds in V.
Using a dynamical approach, we also show the perhaps surprising result that Z² * Z does not embed in V, even though V has many embedded copies of Z² and has many embedded copies of free products of various pairs of its subgroups.