The automorphism tower of groups acting on rooted trees

Abstract:

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 $\operatorname {Aut}^3(\ddot \Gamma )=\operatorname {Aut}^2(\ddot \Gamma )$.