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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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

Said N. Sidki
Affiliation: Universidade de Brasília, Departamento de Matemática, 70.910-900 Brasilia-DF, Brasil

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.