The Automorphism Tower of Groups Acting on Rooted Trees

The group of isometries Aut(Tn) of a rooted n-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group Γ̈ 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 G and Γ̈. We describe this tower for all subgroups of Aut(T2) which decompose as infinitely iterated wreath products. Furthermore, we describe fully the towers of G and Γ̈. More precisely, the tower of 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 Γ̈ has length 2, and its terms are {2, 3}-groups. We show that Aut(Γ̈)/Aut(Γ̈) is an elementary abelian 3-group of countably infinite rank, while Aut (Γ̈) = Aut(Γ̈).