2007/10/18 by Cohen, F. R., Conder, Marston, Lopez, J. +1
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.0710.3515
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 → 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.