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 of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group 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 and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and . More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups. In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while . |