The Diophantine problem in iterated wreath products of free abelian groups is undecidable

In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\dots ,A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in $G$ decides whether or not the system has a solution in $G$.