Wreath products in the automorphism group of a full shift

Abstract

We prove that if a subgroup $H$ of the automorphism group $\Aut(\Sigma^\Z)$ of a non-trivial full shift acts on points of finite support (= points bi-asymptotic to a fixed point) with a free orbit, then for every finitely-generated abelian group $A$, the abstract group $A \wr H$ also embeds in $\Aut(\Sigma^\Z)$. The groups admitting an action with such a free orbit include $A \wr \Z$ for $A$ a finite abelian group, and finitely-generated free groups. The class of such groups is also closed under commensurability and direct products. We obtain for example that $\Z \wr \Z$, $\Z_2 \wr (\Z_2 \wr \Z)$ and $\Z \wr (\Z_2 \wr \Z)$ embed in $\Aut(\Sigma^\Z)$. The group $\Z \wr \Z$ is the first example of a finitely-generated torsion-free subgroup of $\Aut(\Sigma^\Z)$ with infinite cohomological dimension, answering an implicit question of Kim and Roush and an explicit question of the author. We also explore a simpler variant of the construction that gives embeddings of certain Neumann groups, as well as some near-misses to higher iterated wreath products.