  Post's Correspondence Problem (the PCP) is a classical decision problem in
theoretical computer science that asks whether for pairs of free monoid
morphisms $g, h\colon\Sigma^*\to\Delta^*$ there exists any non-trivial
$x\in\Sigma^*$ such that $g(x)=h(x)$.
  Post's Correspondence Problem for a group $\Gamma$ takes pairs of group
homomorphisms $g, h\colon F(\Sigma)\to \Gamma$ instead, and similarly asks
whether there exists an $x$ such that $g(x)=h(x)$ holds for non-elementary
reasons. The restrictions imposed on $x$ in order to get non-elementary
solutions lead to several interpretations of the problem; we consider the
natural restriction asking that $x \notin \ker(g) \cap \ker(h)$ and prove that
the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic
$\Gamma$, but decidable when $\Gamma$ is virtually nilpotent. We also study
this problem for group constructions such as subgroups, direct products and
finite extensions. This problem is equivalent to an interpretation due to
Myasnikov, Nikolev and Ushakov when one map is injective.
