Remarks Concerning Lubotzky's Filtration

TL;DR

This paper develops criteria for when semi-direct product groups are linear, using stable extensions and automorphism tower properties, linking group structure with Lie algebra and hyperplane complement results.

Contribution

It introduces the concept of stable extensions and Galois-like actions to establish linearity of semi-direct products, extending Lubotzky's work.

Findings

Stable extensions ensure the linearity of semi-direct product groups.

Galois-like automorphism actions imply stable extensions and linearity.

Filtration quotients relate to Lie algebras and hyperplane complement groups.

Abstract

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 $1 \to \pi \to G \to \Gamma \to 1$. 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 $\pi$ and $\Gamma$ 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 $\Gamma$ on $\pi$ has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over $\pi$" (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.