Remarks Concerning Lubotzky's Filtration
| dc.creator | Cohen, F. R. | |
| dc.creator | Conder, Marston | |
| dc.creator | Lopez, J. | |
| dc.creator | Prassidis, Stratos | |
| dc.date | 2007-10-18 | |
| dc.date.accessioned | 2026-07-07T08:37:05Z | |
| dc.date.available | 2026-07-07T08:37:05Z | |
| dc.description | 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 $1 \to π\to G \to Γ\to 1$. 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 $π$ 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. | |
| dc.description | 26 pages | |
| dc.identifier | https://arxiv.org/abs/0710.3515 | |
| dc.identifier | http://arxiv.org/abs/0710.3515 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140280 | |
| dc.subject | Group Theory | |
| dc.title | Remarks Concerning Lubotzky's Filtration | |
| dc.type | text |