The Post correspondence problem in groups
Alexei Myasnikov, Andrey Nikolaev, Alexander Ushakov Β· Journal of Group Theory Β· 2014
Abstract We generalize the classical Post correspondence problem (πππ n ) and its non-homogeneous variation (ππππ n ) to non-commutative groups and study the computational complexity of these new problems. We observe that πππ n is closely related to the equalizer problem in groups, while ππππ 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 ππππ n in G in polynomial time. The main results are that πππ n is decidable in a finitely generated nilpotent group in polynomial time, while ππππ 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 πππ and observe that it is in ππ for any group with π-time decidable word problem, meanwhile it is ππ-hard in any group containing free non-abelian subgroups. In particular, the bounded πππ is ππ-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.