Remarks Concerning Lubotzky's Filtration

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A discrete group which admits a faithful, finite dimensional, linear representation over a field 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 → π→ G → Γ→ 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.