Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ automorphisms, and the $Hie(T_n)$ spheromorphisms (hierarchomorphisms), i.~e., homeomorphisms boundary that locally coincide with transformations defined by automorphisms. We show subgroup is spherical in $Hie(T_n)$, any irreducible unitary representation contains at most one $Aut(T_n)$-fixed vector. present a co...

We establish an inequality which involves a non-negative function defined on the vertices of a finite m-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise natura...

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders Tc. This tree only depends on the deformation space of T ; in particular, it is invariant under autom...

This note is intended as an introduction to two previous works respectively by Dahmani, Guirardel, Osin, and Cantat, Lamy. We give proofs of a Small Cancellation Theorem for groups acting on simplicial tree. discuss the application group plane polynomial automorphisms over any ground field.

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 t...

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T c. This tree only depends on the deformation space of T ; in particular, it is invariant under auto...

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications.

Let G = Aut(T ) be the group of automorphisms of a homogeneous tree T and let π be the tensor product of two spherical irreducible unitary representations of G. We complete the explicit decomposition of π commenced in part I of this paper, by describing the discrete series representations of G which appear as subrepresentations of π.

We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of one-relator groups: certain BaumslagSolitar groups and one-relator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.