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 [13] 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 extensionThis 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" [7].(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