A. We use the group (Z2,+) and two associated automorphisms, τ0, τ1, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote T . While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the paren...

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K〈x, y, z〉 over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms ...

We characterize those Lie groups (as well as algebraic groups over a local field of characteristic zero) whose first reduced L-cohomology is zero for all p > 1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field ...

The problem of classifying the automorphisms of a free group of infinite countable rank is investigated. Quite a reasonable generating set for the group AutF∞ is described. Some new subgroups of this group and structural results for them are presented. The main result says that the group of all automorphisms is generated (modulo the IA-automorphisms) by strings and lower triangular automorphisms.

If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.

A group is boundedly simple if for some constant N , every nontrivial conjugacy class generates the whole group in N steps (bounded simplicity implies simplicity). For a large class of trees, Tits proved simplicity of automorphism groups generated by stabilisers of edges. We determine the subclass of trees for which such groups are boundedly simple. This is the class of subdivisions of bi-regul...

We develop a method for detecting symmetries in arbitrary games and exploiting these symmetries when using tree search to play the game. Games in the General Game Playing domain are given as a set of logic based rules defining legal moves, their effects and goals of the players. The presented method transforms the rules of a game into a vertex-labeled graph such that automorphisms of the graph ...

We describe some of the geometric properties of the automorphism group Aut(F ) of Thompson’s group F . We give realizations of Aut(F ) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F ) and their metric properties. In particular, we show that the subg...

This paper gives a classification up to isotopy of automorphisms (self-homeomorphisms) of 3-dimensional handlebodies and compression bodies, analogous to the Nielsen-Thurston classification of automorphisms of surfaces. Indecomposable automorphisms analogous to pseudo-Anosov automorphisms are identified and called generic. The first steps are taken towards understanding generic automorphisms us...