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

TL;DR

This paper introduces the concept of self-similar automata over changing alphabets, demonstrating that all finitely generated residually-finite groups can be represented as such, and constructs automaton models for lamplighter groups over unbounded changing alphabets.

Contribution

It generalizes self-similar groups to automata over changing alphabets and provides automaton representations for lamplighter groups over unbounded alphabets.

Findings

All finitely generated residually-finite groups are self-similar over unbounded changing alphabets.

Constructed automaton models for lamplighter groups with arbitrary finitely generated abelian groups.

Extended the framework of self-similar groups to more general automata settings.

Abstract

Generalizing the idea of self-similar groups defined by Mealy automata, we itroduce 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 $K\wr \mathbb{Z}$ with an arbitrary finitely generated (finite or infinite) abelian group $K$.