Automata groups generated by Cayley machines of groups of nilpotency class two

TL;DR

This paper investigates automata groups generated by Cayley machines of finite nilpotent class two groups, revealing that these automata groups are all cross-wired lamplighters, thus expanding understanding of their structure.

Contribution

It provides explicit presentations of these automata groups and establishes their classification as cross-wired lamplighters, a novel insight into their algebraic structure.

Findings

Automata groups are all cross-wired lamplighters.

Explicit presentations of these groups are provided.

The groups exhibit specific algebraic properties related to nilpotency.

Abstract

We show presentations of automata groups generated by Cayley machines of finite groups of nilpotency class two and these automata groups are all cross-wired lamplighters.