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.

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Metabelian 群：全秩表示、随机性和丢番图问题

我们研究元贝尔群 $\lange A \mid R \rangle_{\mathcal{M}}$ 中元贝尔群的变体 $\mathcal{M}$ 给出的元贝尔群 $G$。我们证明 $G$ 是秩 $\max\{0, |A|-|R|\}$ 的自由元贝尔子群和虚拟阿贝尔正规子群的乘积，并且如果 $|R| \leq |A|-2$ 那么$G$ 的丢番图问题是不可判定的，而如果$|R|\geq |A|$ 是可判定的。我们进一步证明，如果 $|R| \leq |A|-1$ 那么在 $G$ 的任何直接分解中，除了一个，因子实际上都是阿贝尔的。由于有限表示几乎可以肯定地渐近地具有满秩，因此有限表示的元贝尔群几乎可以肯定地渐近地满足所有上述性质。