The automorphism tower of groups acting on rooted trees

The group of isometries $\operatorname{Aut}(\mathcal{T}_n)$ of a rooted $n$-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in $\operatorname{Aut}(\mathcal{T}_n)$. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group $\mathfrak{G}$ studied by R. Grigorchuk, and the group $\ddot\Gamma$ studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as $\mathfrak{G}$ and $\ddot\Gamma$. We describe this tower for all subgroups of $\operatorname{Aut}(\mathcal{T}_2)$ which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of $\mathfrak{G}$ and $\ddot\Gamma$.

More precisely, the tower of $\mathfrak{G}$ is infinite countable, and the terms of the tower are $2$-groups. Quotients of successive terms are infinite elementary abelian $2$-groups.

In contrast, the tower of $\ddot\Gamma$ has length $2$, and its terms are $\{2,3\}$-groups. We show that $\operatorname{Aut}^2(\ddot\Gamma) /\operatorname{Aut}(\ddot\Gamma)$ is an elementary abelian $3$-group of countably infinite rank, while $\operatorname{Aut}^3(\ddot\Gamma)=\operatorname{Aut}^2(\ddot\Gamma)$.