Free products in R. Thompson’s group 𝑉

Abstract

We investigate some product structures in R. Thompson’s group V V , primarily by studying the topological dynamics associated with V V ’s action on the Cantor set C \mathfrak {C} . We draw attention to the class D ( V , C ) \mathcal {D}_{(V,\mathfrak {C})} of groups which have embeddings as demonstrative subgroups of V V whose class can be used to assist in forming various products. Note that D ( V , C ) \mathcal {D}_{(V,\mathfrak {C})} contains all finite groups, the free group on two generators, and Q / Z \mathbf {Q}/\mathbf {Z} , and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G ≤ V G\leq V and H ∈ D ( V , C ) H\in \mathcal {D}_{(V,\mathfrak {C})} , then G ≀ H G\wr H embeds into V V . Finally, if G G , H ∈ D ( V , C ) H\in \mathcal {D}_{(V,\mathfrak {C})} , then G ∗ H G*H embeds in V V .

Using a dynamical approach, we also show the perhaps surprising result that Z 2 ∗ Z Z^2*Z does not embed in V V , even though V V has many embedded copies of Z 2 Z^2 and has many embedded copies of free products of various pairs of its subgroups.