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.