Wreath operations in the group of automorphisms of the binary tree
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.
scientific article; zbMATH DE number 1868054