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The automorphism tower of groups acting on rooted trees
Authors:Laurent Bartholdi   Said N. Sidki
Affiliation:École Polytechnique Fédérale, SB/IGAT/MAD, Bâtiment BCH, 1015 Lausanne, Switzerland ; Universidade de Brasília, Departamento de Matemática, 70.910-900 Brasilia-DF, Brasil
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 $ddotGamma$ 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 $ddotGamma$. 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 $ddotGamma$.

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

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