we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

Let G be a group acting on a tree X such that all edge stabilizers are finite. We extend Bestvina-Handel’s theory of train tracks for automorphisms of free groups to automorphisms of G which permute vertex stabilizers. Using this extension we show that there is an upper bound depending only on G for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms...

We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable. For the existence of automatic automorphisms of word automatic linear orders, we determine the exact level of undecidability in the arithmetical hierarchy.

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have pair of automorphisms.

We define a reflection in a tree as an involutive automorphism whose set of fixed points is a geodesic and prove that, for the case of a homogeneous tree of degree 4k, the topological closure of the group generated by reflections has index 2 in the group of automorphisms of the tree.

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the k-blocks – the maximal verte...

We study properties of eigenvalues a matrix associated with randomly chosen partial automorphism regular rooted tree. show that asymptotically, as the level number goes to infinity, fraction non-zero converges zero in probability.