The automorphism tower of groups acting on rooted trees (original) (raw)
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.
Pro-$$\mathcal {C}$$C congruence properties for groups of rooted tree automorphisms
2018
We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-$$\mathcal {C}$$C completions of the group, where \mathcal {C}$$C is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the \mathcal {C}$$C-congruence subgroup property ($$\mathcal {C}$$C-CSP) if its pro-$$\mathcal {C}$$C completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the \mathcal {C}$$C-CSP. In the case where \mathcal {C}$$C is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.
Groups of tree-automorphisms and their unitary representations
2003
Je tiens à remercier Monsieur le Professeur Marc Burger de m'avoir donné le sujet du travail et de m'avoir été d'une grande aide tout au long de mon travail. Mes remerciements vont aussi à M. le Professeur Pierre de la Harpe, qui a accepté de faire partie du Jury, pour ces remarques et conseils utils.
Groups, Geometry, and Dynamics, 2013
We study the conjugacy problem in the automorphism group Aut(T ) of a regular rooted tree T and in its subgroup 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 ) if and only if they are conjugate in FAut(T ), and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable.
2016
Resum (CAT) L'objectiu d'aquest articlé es discutir i aclarir la noció de fractalitat per a subgrups del grup d'automorfismes d'un arbre arrelat i regular. Per això, definim tres tipus de fractalitat i demostrem, donant contraexemples, que no són equivalents. També presentem alguns resultats que ajuden a determinar el tipus de fractalitat d'un grup donat. Abstract (ENG) 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 counterexamples. Furthermore, we present some tools that are helpful in order to determine the fractality of a given group.
Abelian state-closed subgroups of automorphisms of m-ary trees
2010
The group A m of automorphisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of A m . We prove that the combined diagonal and tree-topological closure A of A is additively a finitely presented Z m OEOEx-module, where Z m is the ring of m-adic integers. Moreover, if A is torsion-free then it is a finitely generated pro-m group. Furthermore, the group A splits over its torsion subgroup. We study in detail the case where A is additively a cyclic Z m OEOEx-module, and we show that when m is a prime number then A is conjugate by a tree automorphism to one of two specific types of groups.