Free products in R. Thompson’s group $V$

Abstract:

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 $\mathfrak {C}$. We draw attention to the class $\mathcal {D}_{(V,\mathfrak {C})}$ of groups which have embeddings as demonstrative subgroups of $V$ whose class can be used to assist in forming various products. Note that $\mathcal {D}_{(V,\mathfrak {C})}$ contains all finite groups, the free group on two generators, and $\mathbf {Q}/\mathbf {Z}$, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If $G\leq V$ and $H\in \mathcal {D}_{(V,\mathfrak {C})}$, then $G\wr H$ embeds into $V$. Finally, if $G$, $H\in \mathcal {D}_{(V,\mathfrak {C})}$, then $G*H$ embeds in $V$.

Using a dynamical approach, we also show the perhaps 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 various pairs of its subgroups.