The automorphism tower of groups acting on rooted trees (original) (raw)

On the automorphism group of a tree

Journal of Combinatorial Theory, Series B, 1977

It is shown that H = T(T), is normal in G = r(T,) for any tree T and any vertex v, if and only if, for all vertices .u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then G/S is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of T, and the structure of such components.

Automorphism groups of trees: generalities and prescribed local actions

New Directions in Locally Compact Groups

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the k-closures of a given group by C. Banks, M. Elder and G. Willis.

Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity

Journal of Mathematical Sciences, 2000

A natural interpretation of automorphisms of one-rooted trees as output automata permits the application of notions of growth and circuit structure in their study. New classes of groups are introduced corresponding to diverse growth functions and circuit structure. In the context of automorphisms of the binary tree, we discuss the structure of maximal 2-subgroups and the question of existence of free subgroups. Moreover, we construct Burnside 2-groups generated by automorphisms of the binary tree which are finite state, bounded, and acyclic.

Affine automorphisms of rooted trees

Geometriae Dedicata, 2016

We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional vector spaces. This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group Z 2 2 ≀ Z.

On the conjugacy problem for finite-state automorphisms of regular rooted trees

2010

We study the conjugacy problem in the automorphism group Aut(T)Aut(T)Aut(T) of a regular rooted tree TTT and in its subgroup FAut(T)FAut(T)FAut(T) of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in Aut(T)Aut(T)Aut(T) if and only if they are conjugate in FAut(T)FAut(T)FAut(T), and that this problem

On the concept of fractality for groups of automorphisms of a regular rooted tree

arXiv: Group Theory, 2016

The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by giving explicit examples. Furthermore we present some tools that are helpful in order to determine the fractality of a given group.