The generalized dihedral groups \(Dih(\mathbb{Z}^n)\) as groups generated by time-varying automata

Abstract

Let \(\mathbb{Z}^n\) be a cubical lattice in the Euclidean space \(\mathbb{R}^n\). The generalized dihedral group  \(Dih(\mathbb{Z}^n)\) is  a topologically discrete group of isometries of \(\mathbb{Z}^n\) generated by  translations and reflections in all points from \(\mathbb{Z}^n\). We study this group as a group generated by a \((2n+2)\)-state time-varying  automaton over the changing alphabet. The corresponding action  on the set of words is described.