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 = Σ 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 Z2 ≀ 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-Aithfully"), we have A ≀ G ≤ X for finite abelian groups A. We show that free products of finite and cyclic groups act with unique moves. We show that Z 2 does not admit move-Aithful actions, and conjecture that Z2 ≀ 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 2V, so it follows that 2V contains all RAAGs, refuting a conjecture of Belk, Bleak and Matucci.

Growth of actions of solvable groups

“…The fact that the group C p Z 2 does not embed into the topological full group of the full shift over Z was conjectured in [Sal21]. We refer to §8.3 for additional applications of our results in this setting.…”

“…By contrast, Theorem 6.14 shows that neither Z Z nor C p Z 2 embed in F pZ, Cq. The case of C p Z 2 solves Conjecture 2 from [Sal21]. More generally for every action of Z d on C, the groups Z Z d and C p Z d`1 do not embed in F pZ d , Cq.…”

On von Neumann regularity of cellular automata

“…We note that the previous construction can also be performed in the topological full group Σ Z of Σ Z for some alphabet Σ (and therefore any nontrivial alphabet [24]), by for a single arithmetic progression defining good runs to consist of maximal powers of the word w = 12 • • • (2n) ∈ {1, . .…”