More on wreath products of cellular automata

Abstract: We prove that if a subgroup $H$ of the automorphism group $\mathrm{Aut}(\Sigma^{\mathbb{Z}})$ of a non-trivial full shift acts on points of finite support with a free orbit, then for every finitely-generated abelian group $A$, the abstract group $A \wr H$ also embeds in $\mathrm{Aut}(\Sigma^{\mathbb{Z}})$. The groups admitting an action with such a free orbit include $A \wr {\mathbb{Z}}$ 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 ${\mathbb{Z}} \wr {\mathbb{Z}}$, ${\mathbb{Z}}_2 \wr ({\mathbb{Z}}_2 \wr {\mathbb{Z}})$ and ${\mathbb{Z}} \wr ({\mathbb{Z}}_2 \wr {\mathbb{Z}})$ embed in $\mathrm{Aut}(\Sigma^{\mathbb{Z}})$. To our knowledge, the group ${\mathbb{Z}} \wr {\mathbb{Z}}$ is the first example of a finitely-generated torsion-free subgroup of $\mathrm{Aut}(\Sigma^{\mathbb{Z}})$ with infinite cohomological dimension. It answers 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 some near-misses to iterated permutational wreath products, as well as some Neumann groups.