Metabelian groups: full-rank presentations, randomness and Diophantine problems

We study metabelian groups $G$ given by full rank finite presentations $⟨A\mid R{⟩}_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max \{0,\mathrm{\mid }A\mathrm{\mid }-\mathrm{\mid }R\mathrm{\mid }\}$ and a virtually abelian normal subgroup, and that if $\mathrm{\mid }R\mathrm{\mid }\le \mathrm{\mid }A\mathrm{\mid }-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $\mathrm{\mid }R\mathrm{\mid }\ge \mathrm{\mid }A\mathrm{\mid }$. We further prove that if $\mathrm{\mid }R\mathrm{\mid }\le \mathrm{\mid }A\mathrm{\mid }-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.