More on Groups and Counter Automata

TL;DR

This paper provides a new combinatorial proof linking automata acceptance of group word problems to the group's algebraic structure, specifically showing that certain automaton acceptances imply the group is virtually abelian and establishing a homomorphism between the groups.

Contribution

It offers an elementary, combinatorial proof of a known theorem and explicitly constructs a group homomorphism connecting the automaton's group and the finitely generated group.

Findings

Automaton acceptance implies the group is virtually abelian.

Explicit homomorphism from a subgroup of G to a finite index subgroup of H.

Elementary combinatorial proof of the theorem.

Abstract

Elder, Kambites, and Ostheimer showed that if the word problem of a finitely generated group $H$ is accepted by a $G$-automaton for an abelian group $G$, then $H$ is virtually abelian. We give a new, elementary, and purely combinatorial proof to the theorem. Furthermore, our method extracts an explicit connection between the two groups $G$ and $H$ from the automaton as a group homomorphism from a subgroup of $G$ onto a finite index subgroup of $H$.