论文标题
Metabelian群体：全等级演示，随机性和二芬太汀问题
Metabelian groups: full-rank presentations, randomness and Diophantine problems
论文作者
论文摘要
我们研究Metabelian组的Mathabelian $ g $ $ g $由全等级有限演示文稿$ \ langle a \ langle a \ mid rangle _ {\ Mathcal {m}} $我们证明，$ g $是$ \ max \ {0，| a | - | - | r | \} $和几乎是Abelian正常子组的产品的产物，如果$ | r | \ leq | a | -2 $，那么$ g $的二芬太丁问题是不确定的，而如果$ | r | r | \ geq | a | $可以决定。我们进一步证明，如果$ | r | \ leq | a | -1 $，然后在$ g $的任何直接分解中，但一个因素实际上是阿贝利安。由于有限的演示文稿几乎肯定地逐渐渐近，因此几乎可以肯定地毫无疑问地呈现上述所有特性。
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.
