The graph structure of graph groups that are subgroups of Thompson's group $V$

TL;DR

This paper characterizes which graph products, specifically Right Angled Artin Groups, embed into Thompson's group V, revealing that the only obstruction is the presence of a Z^2 * Z subgroup, and provides a graph-theoretic criterion for embeddability.

Contribution

It establishes a complete characterization of embeddability of graph products into V, identifying the sole obstruction and offering a new graph-theoretic perspective.

Findings

Z^2 * Z is the only obstruction to embedding.

Provides a graph-theoretic criterion for embeddability.

Characterizes subgroups of Thompson's group V.

Abstract

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson's group $V$. It was shown by Bleak and Salazar-Diaz that $\mathbb{Z}^2 * \mathbb{Z}$ was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.