Lamplighter groups and automata

This paper investigates presentations of lamplighter groups using computational models from automata theory. The present work shows that if G can be presented such that the full group operation is recognised by a transducer, then the same is true for the lamplighter group \(G \wr {{\mathbb {Z}}}\) of G. Furthermore, Cayley presentations, where only multiplications with constants are recognised by transducers, are used to study generalised lamplighter groups of the form \(G \wr {{\mathbb {Z}}}^d\) and \(G \wr F_d\), where \(F_d\) is the free group over d generators. Additionally, \({{\mathbb {Z}}}_k \wr {{\mathbb {Z}}}^2\) and \({{\mathbb {Z}}}_k \wr {F_d}\) are shown to be Cayley tree automatic.