论文标题
在方程式上的Noetherian和残留有限的基团
On equationally Noetherian and residually finite groups
论文作者
论文摘要
本文的目的是将剩余组的类别类别与方程式的noetherian群体进行比较和对比 - 每个无系数方程系统都等同于有限子系统的组。构造剩余有限但不方程式的组（例如，所有有限组的直接总和），反之亦然的组，反之亦然（例如，添加群$（\ Mathbb {q}，+）$）。但是，在文献中似乎没有出现有限生成的明确示例。 在本文中，我们表明，在有限生成的群体中，残留有限和方程式的Noetherian组类似，但它们都不包含另一个。一方面，我们表明某些有限生成的群体被称为残留有限的群体，例如Abelian by-polycyclic群体，也是方程式的Noetherian（回答了R. Bryant提出的问题）。我们还给出了类似的结果，表明了足够的条件，使组的基本组是方程式noetherian和剩余有限的条件。另一方面，我们产生了有限产生的非（方程式noetherian）组的示例，这些示例是无残留的无扭转的nilpotent或可分离性的，以及有限提出的方程式noetherian群体的示例，这些noetherian群体不是残留有限的。
The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It is easy to construct groups that are residually finite but not equationally Noetherian (e.g. the direct sum of all finite groups) or vice versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no explicit examples that are finitely generated seem to appear in the literature. In this paper, we show that among finitely generated groups, the classes of residually finite and equationally Noetherian groups are similar, but neither of them contains the other. On one hand, we show that some classes of finitely generated groups which are known to be residually finite, such as abelian-by-polycyclic groups, are also equationally Noetherian (answering a question posed by R. Bryant). We also give analogous results stating sufficient conditions for a fundamental group of a graph of groups to be equationally Noetherian and to be residually finite. On the other hand, we produce examples of finitely generated non-(equationally Noetherian) groups which are either residually torsion-free nilpotent or conjugacy separable, as well as examples of finitely presented equationally Noetherian groups that are not residually finite.
