Free products in R. Thompson's group V

TL;DR

This paper explores the embedding of various free product structures into Thompson's group V, revealing which free products can be embedded and highlighting surprising non-embeddings despite V's rich subgroup structure.

Contribution

It identifies specific free products that embed into V and demonstrates a surprising non-embedding result, advancing understanding of V's subgroup structure.

Findings

Free product of any two finite groups embeds in V

Free product of any finite group with Q/Z embeds in V

Countable non-abelian free groups embed in V

Abstract

We investigate free product structures in R. Thompson's group V, primarily by studying the topological dynamics associated with V's action on the Cantor Set. We show that the class of free products which can be embedded into V includes the free product of any two finite groups, the free product of any finite group with Q/Z, and the countable non-abelian free groups. We also show the somewhat 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 pairs of its subgroups.