Post's Correspondence Problem for hyperbolic and virtually nilpotent groups

Abstract

AbstractPost's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms there exists any non‐trivial such that . PCP for a group takes pairs of group homomorphisms instead, and similarly asks whether there exists an such that holds for non‐elementary reasons. The restrictions imposed on in order to get non‐elementary solutions lead to several interpretations of the problem; we mainly consider the natural restriction asking that and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic , but decidable when is virtually nilpotent, en route also studying this problem for finite extensions. We also consider a different interpretation of the PCP due to Myasnikov, Nikolaev and Ushakov, proving decidability for torsion‐free nilpotent groups.