The automorphism tower of groups acting on rooted trees

The group of isometries Aut(T n ) of a rooted n-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(T n ). 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 F 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 Γ. We describe this tower for all subgroups of Aut(T 2 ) which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of & and r. More precisely, the tower of & 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 F has length 2, and its terms are {2,3}-groups. We show that Aut 2 (Γ)/ Aut(Γ) is an elementary abelian 3-group of countably infinite rank, while Aut 3 (Γ) = Aut 2 (Γ).