The Post correspondence problem in groups

Abstract

Abstract We generalize the classical Post correspondence problem (𝐏𝐂𝐏 n ) and its non-homogeneous variation (𝐍𝐏𝐂𝐏 n ) to non-commutative groups and study the computational complexity of these new problems. We observe that 𝐏𝐂𝐏 n is closely related to the equalizer problem in groups, while 𝐍𝐏𝐂𝐏 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 hereditary word problem) can be reduced to 𝐍𝐏𝐂𝐏 n in G in polynomial time. The main results are that 𝐏𝐂𝐏 n is decidable in a finitely generated nilpotent group in polynomial time, while 𝐍𝐏𝐂𝐏 n is undecidable in any group containing free non-abelian subgroups (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 𝐏𝐂𝐏 and observe that it is in 𝐍𝐏 for any group with 𝐏-time decidable word problem, meanwhile it is 𝐍𝐏-hard in any group containing free non-abelian subgroups. In particular, the bounded 𝐏𝐂𝐏 is 𝐍𝐏-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.