Matthews, C. R.; Vaserstein, L. N.; Weisfeiler, Boris Congruence properties of Zariski-dense subgroups. I. (English) Zbl 0551.20029 Proc. Lond. Math. Soc., III. Ser. 48, 514-532 (1984). The authors prove the following Theorem. Let G be a connected simply connected absolutely almost simple algebraic group defined over \({\mathbb{Q}}\) and let \(\Gamma\) be a finitely generated subgroup in G(\({\mathbb{Q}})\) which is Zariski-dense in G. Then for all sufficiently large prime numbers p the reduction \(\Gamma_ p\) of \(\Gamma\) coincides withe \(G_ p({\mathbb{F}}_ p)\). They give also applications of this theorem to the study of properties of p-adic and adelic closures of various subgroups in G. Reviewer: S.I.Gel’fand Cited in 2 ReviewsCited in 51 Documents MSC: 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth Keywords:connected simply connected absolutely almost simple algebraic group; finitely generated subgroup; Zariski-dense; reduction; adelic closures PDFBibTeX XMLCite \textit{C. R. Matthews} et al., Proc. Lond. Math. Soc. (3) 48, 514--532 (1984; Zbl 0551.20029) Full Text: DOI