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.
Categories: math.GR math.DS