Groups Whose Word Problems Are Accepted by Abelian $$G$$-Automata

Abstract

Elder, Kambites, and Ostheimer showed that if a finitely generated group $$H$$ has word problem accepted by a $$G$$ -automaton for an abelian group $$G$$ , then $$H$$ has an abelian subgroup of finite index. Their proof is, however, non-constructive in the sense that it is by contradiction: they proved that $$H$$ must have a finite index abelian subgroup without constructing any finite index abelian subgroup of $$H$$ . In addition, a part of their proof is in terms of geometric group theory, which makes it hard to read without knowledge of the field. We give a new, elementary, and in some sense more constructive proof of the theorem, in which we construct, from the abelian $$G$$ -automaton accepting the word problem of $$H$$ , a group homomorphism from a subgroup of $$G$$ onto a finite index subgroup of $$H$$ . Our method is purely combinatorial and contains no geometric arguments.