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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The automorphism tower of groups acting on rooted trees
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by Laurent Bartholdi and Said N. Sidki PDF
Trans. Amer. Math. Soc. 358 (2006), 329-358 Request permission


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:
  • Said N. Sidki
  • Affiliation: Universidade de Brasília, Departamento de Matemática, 70.910-900 Brasilia-DF, Brasil
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  • 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”.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 329-358
  • MSC (2000): Primary 20F28; Secondary 20E08
  • DOI:
  • MathSciNet review: 2171236