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

TL;DR

This paper investigates the structure and decision problems of finitely presented metabelian groups, revealing conditions for their decomposition, decidability of Diophantine problems, and typical properties in the asymptotic sense.

Contribution

It provides a detailed structural analysis of metabelian groups with full-rank presentations, establishing new results on their decompositions and the decidability of their Diophantine problems.

Findings

G is a product of a free metabelian subgroup and a virtually abelian subgroup.

Diophantine problem is undecidable if |R| ≤ |A|-2, decidable if |R| ≥ |A|.

Finitely presented metabelian groups are asymptotically almost surely full rank and satisfy the properties discussed.

Abstract

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$ 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.