Group equations with abelian predicates

TL;DR

This paper investigates the decidability of group equations with abelian predicates, extending previous work on word equations, and identifies conditions under which the problem is undecidable or decidable in various classes of groups.

Contribution

It introduces a systematic study of equations with abelian predicates in groups, establishing undecidability results for certain groups and decidability in others based on their algebraic properties.

Findings

Undecidability of equations with abelian predicates in right-angled Artin groups and certain hyperbolic groups.

Decidability of the problem in groups with finite abelianisation, including right-angled Coxeter groups and hyperbolic groups with finite abelianisation.

Abstract

In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups, and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation.