Cross-wired lamplighter groups and linearity of automata groups

TL;DR

This paper explores the structure of automata groups derived from certain nilpotent groups, revealing their classification as cross-wired lamplighter groups and establishing conditions for their linearity.

Contribution

It introduces a presentation for automata groups from Cayley machines of finite step two nilpotent groups and proves their linearity in specific cases.

Findings

Automata groups are cross-wired lamplighter groups.

These groups do not embed in wreath products of finite and torsion-free groups.

Certain automata groups are proven to be linear.

Abstract

We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machine and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a cross-wired lamplighter group and does not embed in the wreath product of a finite group with a torsion free group. For a subfamily of such finite step two nilpotent groups, we prove that their associated automata groups are linear.