Undecidability of the submonoid membership problem for free nilpotent group of class $l\geqslant 2$ of sufficiently large rank

Abstract

An answer is given to the question of M. Lohrey and B. Steinberg on decidability of the submonoid membership problem for a finitely generated nilpotent group. Namely, a finitely generated submonoid of a free nilpotent group of class $2$ of sufficiently large rank $r$ is constructed, for which the membership problem is algorithmically undecidable. This implies the existence of a submonoid with similar property in any free nilpotent group of class $l \geqslant 2$ of rank $r$. The proof is based on the undecidability of Hilbert's tenth problem.