  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.
