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Congruence properties of Zariski-dense subgroups. I. (English) Zbl 0551.20029
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

##### 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
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