亚阿贝尔群：满秩表示，随机性和丢番图问题
Metabelian groups: full-rank presentations, randomness and Diophantine problems
摘要：
研究了亚阿贝尔群的簇$\mathcal{M}$中满秩有限表示$\langle A \mid R \rangle_{\mathcal{M}}$给出的亚阿贝尔群$G$.证明了$G$是秩为$\max\{0, |A|-|R|\}$的自由亚阿贝尔子群与虚阿贝尔正规子群的乘积，若$|R| \leq |A|-2$，则$G$的丢番图问题不可判定，若$|R|\geq |A|$，则$G$的丢番图问题可判定.我们进一步证明，如果$|R| \leq |A|-1$，然后在任何直接分解的$G$的所有，但一个，因素几乎是阿贝尔。由于有限表示几乎必然渐近满秩，所以有限表示亚阿贝尔群几乎必然渐近满足上述所有性质。
英文摘要：
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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