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The automorphism tower of groups acting on rooted trees


Authors: Laurent Bartholdi and Said N. Sidki
Journal: Trans. Amer. Math. Soc. 358 (2006), 329-358
MSC (2000): Primary 20F28; Secondary 20E08
DOI: https://doi.org/10.1090/S0002-9947-05-03712-8
Published electronically: March 31, 2005
MathSciNet review: 2171236
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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 $\operatorname{Aut}^3(\ddot\Gamma)=\operatorname{Aut}^2(\ddot\Gamma)$.


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Additional Information

Laurent Bartholdi
Affiliation: École Polytechnique Fédérale, SB/IGAT/MAD, Bâtiment BCH, 1015 Lausanne, Switzerland
Email: laurent.bartholdi@epfl.ch

Said N. Sidki
Affiliation: Universidade de Brasília, Departamento de Matemática, 70.910-900 Brasilia-DF, Brasil
Email: sidki@mat.unb.br

DOI: https://doi.org/10.1090/S0002-9947-05-03712-8
Received by editor(s): August 15, 2003
Received by editor(s) in revised form: March 12, 2004
Published electronically: March 31, 2005
Additional Notes: The authors gratefully acknowledge support from the “Fonds National Suisse de la Recherche Scientifique”.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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