On finite and elementary generation of SL_2(R)
| dc.creator | Abramenko, Peter | |
| dc.date | 2008-08-07 | |
| dc.date.accessioned | 2026-07-07T09:55:27Z | |
| dc.date.available | 2026-07-07T09:55:27Z | |
| dc.description | Motivated by a question of A. Rapinchuk concerning general reductive groups, we are investigating the following question: Given a finitely generated integral domain $R$ with field of fractions $F$, is there a \emph{finitely generated subgroup} $Γ$ of $SL_2(F)$ containing $SL_2(R)$? We shall show in this paper that the answer to this question is negative for any polynomial ring $R$ of the form $R = R_0[s,t]$, where $R_0$ is a finitely generated integral domain with infinitely many (non--associate) prime elements. The proof applies Bass--Serre theory and reduces to analyzing which elements of $SL_2(R)$ can be generated by elementary matrices with entries in a given finitely generated $R$--subalgbra of $F$. Using Bass--Serre theory, we can also exhibit new classes of rings which do not have the $GE_2$ property introduced by P.M. Cohn. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/0808.1095 | |
| dc.identifier | http://arxiv.org/abs/0808.1095 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/166633 | |
| dc.subject | Group Theory | |
| dc.subject | 20E06, 20E08, 20H05 | |
| dc.title | On finite and elementary generation of SL_2(R) | |
| dc.type | text |