{
  "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 $\\operatorname {Aut}^3(\\ddot \\Gamma )=\\operatorname {Aut}^2(\\ddot \\Gamma )$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "math/0308127",
  "author_total": 2,
  "authors": [
    {
      "name": "Laurent Bartholdi"
    },
    {
      "name": "Said N. Sidki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0z3283dws6nkt.bib",
  "cited_by": 10,
  "doi": "10.1090/s0002-9947-05-03712-8",
  "fulltext_url": null,
  "id": "0z3283dws6nkt",
  "issns": [
    "0002-9947",
    "1088-6850"
  ],
  "landing_url": "https://hal.science/hal-05849569v1",
  "language": "en",
  "licence": "PDM-1.0",
  "licence_label": "Public domain",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/publicdomain/mark/1.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-gr"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W1801770975",
  "pdf_url": "https://www.ams.org/tran/2006-358-01/S0002-9947-05-03712-8/S0002-9947-05-03712-8.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": "American Mathematical Society",
  "ris": "https://research.draconitic.com/paper/0z3283dws6nkt.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:18Z",
      "native_id": "hal-05849569",
      "source": "hal",
      "url": "https://hal.science/hal-05849569v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-gr",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The automorphism tower of groups acting on rooted trees",
  "title_text": "The automorphism tower of groups acting on rooted trees",
  "type": "peer-reviewed",
  "updated_at": "2026-11-05T12:20:12Z",
  "url": "https://research.draconitic.com/paper/0z3283dws6nkt",
  "venue": "Transactions of the American Mathematical Society",
  "versions": []
}