Graph and wreath products in topological full groups of full shifts

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.