Post's correspondence problem for hyperbolic and virtually nilpotent groups

TL;DR

This paper investigates the decidability of Post's Correspondence Problem within hyperbolic and virtually nilpotent groups, showing undecidability in the former and decidability in the latter, with implications for group constructions.

Contribution

It establishes the decidability status of a restricted PCP in hyperbolic and virtually nilpotent groups, extending understanding of computational problems in geometric group theory.

Findings

Undecidable for hyperbolic groups

Decidable for virtually nilpotent groups

Analyzes subgroup, direct product, and finite extension cases

Abstract

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.