关于Lubotzky过滤的注记
Remarks Concerning Lubotzky's Filtration
摘要：
在特征为零的域$\mathbb F$上有一个忠实的有限维线性表示的离散群称为线性群。本文将半直积的自然结构与A. Lubotzky研究线性表示的存在性，以开发一种技术来给出证明半直积是线性的充分条件。设$G$表示由分裂扩张$1 \to π\to G \to Γ\to 1$给出的半直积的离散群.本文定义了这种半直积的另一种结构类型，称为下面的稳定扩张。主要研究结果如下：1.如果$π$和$Γ$是线性的，并且扩展是稳定的，则$G$也是线性的。关于该扩展的限制是必要的，以保证$G$是线性的，如从Formanek-Procesi“毒物基团”的性质所见。2.如果$Γ$对$π$的作用具有“类伽罗瓦”性质，即它通过某些自然的“$π$上的群塔“（将在下面定义）的自同构进行因子分解，则相关的扩张是稳定的，因此$G$是线性的。3.一个稳定的扩展的条件也意味着$G$承认过滤子本身给一个自然的结构李代数，这也意味着早期的结果Kohno，和Falk-Randell的李代数重视下降中心列相关联的基本群的复超平面补。这里的方法表明，一个可能的技术，获得新的线性结果可能是分析自同构的塔组。
英文摘要：
A discrete group which admits a faithful, finite dimensional, linear representation over a field $\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear. Let $G$ denote a discrete group which is a semi-direct product given by a split extension $1 \to π\to G \to Γ\to 1$. This note defines an additional type of structure for this semi-direct product called a stable extension below. The main results are as follows: 1. If $π$ and $Γ$ are linear, and the extension is stable, then $G$ is also linear. Restrictions concerning this extension are necessary to guarantee that $G$ is linear as seen from properties of the Formanek-Procesi "poison group". 2. If the action of $Γ$ on $π$ has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over $π$" (to be defined below), then the associated extension is stable and thus $G$ is linear. 3. The condition of a stable extension also implies that $G$ admits filtration quotients which themselves give a natural structure of Lie algebra and which also imply earlier results of Kohno, and Falk-Randell on the Lie algebra attached to the descending central series associated to the fundamental groups of complex hyperplane complements. The methods here suggest that a possible technique for obtaining new linearity results may be to analyze automorphisms of towers of groups.
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