The n-ary adding machine and solvable groups

We describe under various conditions abelian subgroups of the automorphism‎ ‎group \mathrm{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 \left( ۰,۱‎, ‎\dots‎, ‎n-۱\right) ‎. ‎As‎ ‎an application‎, ‎for n=p a prime number‎, ‎and for n=۴‎, ‎we prove that‎ ‎every soluble subgroup of \mathrm{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