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.

翻译：Post对应问题（PCP）是理论计算机科学中的经典判定问题，询问是否存在自由幺半群同态对$g, h\colon\Sigma^*\to\Delta^*$及非平凡元素$x\in\Sigma^*$使得$g(x)=h(x)$。对于群$\Gamma$的Post对应问题则考虑群同态对$g, h\colon F(\Sigma)\to \Gamma$，类似地询问是否存在$x$使得$g(x)=h(x)$成立且非平凡。为得到非平凡解而对$x$施加的约束导致了该问题的多种解释；我们考虑自然约束$x \notin \ker(g) \cap \ker(h)$，并证明该PCP解释在任意双曲群$\Gamma$上不可判定，但在$\Gamma$为殆幂零群时可判定。我们还研究了子群、直积与有限扩张等群构造下的该问题。当其中一个映射为单射时，该问题等价于Myasnikov、Nikolev与Ushakov提出的解释。