The Post correspondence problem in groups

Abstract

We generalize the classical Post correspondence problem ($\mathbf{PCP}_n$) and its non-homogeneous variation ($\mathbf{GPCP}_n$) to non-commutative groups and study the computational complexity of these new problems. We observe that $\mathbf{PCP}_n$ is closely related to the equalizer problem in groups, while $\mathbf{GPCP}_n$ is connected to the double twisted conjugacy problem for endomorphisms. Furthermore, it is shown that one of the strongest forms of the word problem in a group $G$ (we call it the {\em hereditary word problem}) can be reduced to $\mathbf{GPCP}_n$ in $G$ in polynomial time. The main results are that $\mathbf{PCP}_n$ is decidable in a finitely generated nilpotent group in polynomial time, while $\mathbf{GPCP}_n$ is undecidable in any group containing free non-abelian subgroup (though the argument is very different from the classical case of free semigroups). We show that the double endomorphism twisted conjugacy problem is undecidable in free groups of sufficiently large finite rank. We also consider the bounded $\mathbf{PCP}$ and observe that it is in $\mathbf{NP}$ for any group with $\mathbf{P}$-time decidable word problem, meanwhile it is $\mathbf{NP}$-hard in any group containing free non-abelian subgroup. In particular, the bounded $\mathbf{PCP}$ is $\mathbf{NP}$-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.