Wreath operations in the group of automorphisms of the binary tree
type: Item
instance of: scholarly article
title: Wreath operations in the group of automorphisms of the binary tree
zbMATH Open document ID: 1027.20018
DOI: 10.1016/S0021-8693(02)00050-9
author: Yanyan Li
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The group \({\mathcal F}_n\) of finite-state automorphisms is defined to be the enumerable group of automorphisms which correspond to the finite-state input-output automata on the alphabet \(\{0,1,\dots,n-1\}\), where automata are a natural interpretation of the automorphisms of the \(n\)-ary one-rooted regular tree. The current paper studies the group \(\mathcal A\) of automorphisms of the binary tree and, especially, of its subgroup \(\mathcal F\) consisting of finite-state automorphisms. An operation called `tree-wreathing' is defined on \(\mathcal A\). For a given subgroup \(H\) of \(\mathcal A\) and a free Abelian group \(K\) of finite rank \(r\) this new operation produces uniformly copies in \(\mathcal A\) such that the group \(G=H\overline\wr K\) generated by them is an overgroup of the restricted wreath product \(H\wr K\). Moreover, \(G\) contains the infinite direct sum \(N\) of copies of the derived group \(H'\), and \(G/N\) is isomorphic to \(H\wr K\). This new operation preserves solvability, torsion-freeness, and having finite state. A faithful representation of any free metabelian group of finite rank is obtained as a finite-state group of automorphisms of the binary tree.
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 1868054
zbMATH Keywords: wreath products
zbMATH Keywords: rooted binary trees
zbMATH Keywords: finite state automata
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: finite state automorphisms
zbMATH Keywords: free metabelian groups
zbMATH Keywords: faithful representations
reviewed by: Vahagn H. Mikaelian
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00050-9
OpenAlex ID: W2047531200
description: scientific article; zbMATH DE number 1868054