Free products in R. Thompson's group V

We investigate some product structures in R. Thompson's group V's action on the Cantor set \mathfrak{C} \mathcal {D}_{(V,\mathfrak{C})}demonstrative subgroups of V whose class can be used to assist in forming various products. Note that \mathcal {D}_{(V,\mathfrak{C})} \mathbf {Q}/\mathbf {Z} and H\in \mathcal {D}_{(V,\mathfrak{C})} embeds into V, H\in \mathcal {D}_{(V,\mathfrak{C})} embeds in V Using a dynamical approach, we also show the perhaps surprising result that Z^2*Z, even though V and has many embedded copies of free products of various pairs of its subgroups.