The concept of self-similar automata over a changing alphabet and lamplighter groups generated by such automata

Generalizing the idea of self-similar groups defined by Mealy automata, we introduce the notion of a self-similar automaton and a self-similar group over a changing alphabet. We show that every finitely generated residually-finite group is self-similar over an arbitrary unbounded changing alphabet. We construct some naturally defined self-similar automaton representations over an unbounded changing alphabet for any lamplighter group K≀Z with an arbitrary finitely generated (finite or infinite) abelian group K.