The lamplighter group ${\mathbb{Z}}_3\wr \mathbb{Z}$ generated by a bireversible automaton

I. Bondarenko, D. D'Angeli, E. Rodaro

We construct a bireversible self-dual automaton with $3$ states over an alphabet with $3$ letters which generates the lamplighter group ${\mathbb{Z}}_3\wr \mathbb{Z}$.