The n-ary adding machine and solvable groups

Abstract

We describe under a various conditions abelian subgroups of the automorphism group $Aut(T_n)$ of the regular $n$-ary tree $T_n$, which are normalized by the $n$-ary adding machine $tau=(e,dots, e,tau)sigma_tau$ where $sigma_tau$ is the $n$-cycle $(0, 1,dots, n-1)$. As an application, for $n=p$ a prime number, and for $n=p^2$ when $p=2$, we prove that every finitely generated soluble subgroup of $Aut(T_n)$, containing $tau$ is an extension of a torsion-free metabelian group by a finite group.

Keywords

• Adding machine
• Tree automorphisms
• Automata
• Solvable Groups