We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (“Mother groups”) which is done by analyzing the asymptotic properties of random walks on these groups.

We construct tree-decompositions of graphs that distinguish all their k-blocks and tangles of order k, for any fixed integer k. We describe a family of algorithms to construct such decompositions, seeking to maximize their diversity subject to the requirement that they commute with graph isomorphisms. In particular, all the decompositions constructed are invariant under the automorphisms of the...

A constant spinal group is a subgroup of the automorphism regular rooted tree, generated by automorphisms and directed B whose action on subtree equal to global action. We provide two conditions in terms certain dynamical systems determined for groups be periodic, generalising previous results Grigorchuk–Gupta–Sidki other related constructions. This allows us various new examples finitely infin...

We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, es...

let r and s be reduced rings with identities whose idempotents are central, and let m be an(r, s)-bimodule such that annr (m)=0. in this paper, we determine first the structure of automorphisms ofthe triangular ring, =sr mt0, and then, for all automorphisms α ,β of t we determine the structureof (α,β)-derivations of t.

This is the second paper dedicated to a detailed study of the infinite 3-group ?J= (74 r>, 1’3 = rr;3 = 1, y=(y, CT, 2-l) (a recursive definition to be explained later), which was introduced by the author and Narain Gupta in [2] as a group of automorphisms of an infinite regular tree F. As was shown in [4], Y has the remarkable properties of containing a copy of every finite 3-group, and having...

We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions on...

Let G = Aut(T ) be the group of automorphisms of a homogeneous tree T , and let Γ be a lattice subgroup of G. Let π be the tensor product of two spherical irreducible unitary representations of G. We give an explicit decomposition of the restriction of π to Γ. We also describe the spherical component of π explicitly, and this decomposition is interpreted as a multiplication formula for associat...