Graph and wreath products in topological full groups of full shifts

TL;DR

This paper demonstrates that topological full groups of full shifts contain all right-angled Artin groups and many other complex groups, revealing their rich subgroup structure and embedding properties, including connections to Thompson groups.

Contribution

It proves that topological full groups of full shifts contain all RAAGs and many other groups, and explores their subgroup closure properties and embeddings, refuting previous conjectures.

Findings

Topological full groups of full shifts contain every RAAG.

The subgroup family with linear look-ahead is closed under graph products.

The group 2V contains all RAAGs, refuting a prior conjecture.

Abstract

We prove that the topological full group $[[X]]$ of a two-sided full shift $X = \Sigma^{\mathbb{Z}}$ contains every right-angled Artin group (also called a graph group). More generally, we show that the family of subgroups with "linear look-ahead" is closed under graph products. We show that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ embeds in $[[X]]$, and conjecture that it does not embed in $[[X]]$ with linear look-ahead. Generalizing the lamplighter group, we show that whenever $G$ acts with "unique moves" (or at least "move-$A$ithfully"), we have $A \wr G \leq [[X]]$ for finite abelian groups $A$. We show that free products of finite and cyclic groups act with unique moves. We show that $\mathbb{Z}^2$ does not admit move-$A$ithful actions, and conjecture that $\mathbb{Z}_2 \wr \mathbb{Z}^2$ does not embed in $[[X]]$ at all. We show that topological full groups of all infinite nonwandering sofic shifts have the same subgroups, and that this set of groups is closed under commensurability. The group $[[X]]$ embeds in the higher-dimensional Thompson group $2$V, so it follows that $2$V contains all RAAGs, refuting a conjecture of Belk, Bleak and Matucci.