  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.
