The lamplighter group of rank two generated by a bireversible automaton

TL;DR

This paper constructs a specific bireversible automaton with four states that generates the rank-two lamplighter group, linking automaton actions to affine transformations on formal power series.

Contribution

It introduces a new 4-state bireversible automaton that generates the rank-two lamplighter group, connecting automaton theory with algebraic structures.

Findings

Automaton generates the lamplighter group of rank two.

Automaton action corresponds to affine transformations.

Provides a new automaton model for the group.

Abstract

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $(\mathbb Z_2^2)\wr\mathbb Z$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring $\mathbb Z_2[[t]]$ of formal power series over $\mathbb Z_2$.