More on Groups and Counter Automata
Takao Yuyama

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 is accepted by a -automaton for an abelian group , then 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 and from the automaton as a group homomorphism from a subgroup of onto a finite index subgroup of .
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Taxonomy
Topicssemigroups and automata theory · Geometric and Algebraic Topology · Finite Group Theory Research
