On solvability of regular equations in the variety of metabelian groups

We study the solvability of equations over groups within a given variety or another class of groups. The classes of nilpotent and solvable groups were considered as main classes to investigate from such point of view. The natural analogues of the famous Kervaire–Laudenbach and Levin conjectures were raised to the challenge. It was also noted that the “solvable” version of the known theorem by Brodski\v i is not true. In this paper, for each $n\in\mathbb N$, $n\geq2$, we prove that every regular equation over the free metabelian group $M_n$ is solvable in the class $\mathcal M$ of all metabelian groups. Moreover, there is a metabelian group $\tilde{M}_n$ that contains a solution of every unimodular equation over $M_n$. These results are extended to the class of rigid metabelian groups. Also, we give an example showing that there exists an equation over a locally indicable torsion-free metabelian group $G$ that has no solution in any solvable overgroup of $G$. It follows that solvable versions of the Levin conjecture are not true. Another example presents an unimodular equation over a locally indicable torsion-free metabelian group $G$ that has no solution in any metabelian overgroup of $G$. Hence, the Kervaire–Laudenbach conjecture is not valid for the variety of all metabelian groups. We prove that there is an unimodular equation over a finite metabelian group $G$ that has no solutions in any finite metabelian overgroup of $G$. This means that analog of the famous theorem by Gerstenhaber and Rothaus (about solvability of each unimodular equation over a finite group $G$ in some finite overgroup of $G$) is not valid for the class of finite metabelian groups.