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Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups. (English) Zbl 0568.14025
In this paper the author gives “approximations” for Zariski-dense subgroups of semi-simple algebraic groups. To make it precise, let $$G$$ be an almost simple, connected and simply-connected algebraic group over an algebraically closed field of characteristic different from 2 and 3. Let $$\Gamma$$ be a finitely generated Zariski-dense subgroup of $$G(k)$$. Denoting the adjoint action of $$G$$ on $$\mathrm{Lie}\, G$$, the Lie algebra of $$G$$, by Ad, let $$A$$ be the subring of $$k$$ generated by the traces $$\text{tr Ad}\,\gamma$$, $$\gamma\in\Gamma$$. The author shows that $$G$$ has a structure $$G_A$$ of a group scheme over $$A$$ and for a suitable choice of $$b$$ in $$A$$, the author shows that $$A_b$$ is an affine $$k$$-algebra and regular, and $$G_{A_b}(A_b)\cap \Gamma$$ is of finite index in $$\Gamma$$. The author exhibits a certain normal subgroup $$\Gamma'$$ of $$\Gamma$$ contained in $$G_{A_b}(A_ b)\cap \Gamma$$ such that the reduction of $$\Gamma'$$ modulo most maximal ideals $$M$$ of $$A_b$$ is $$G_{A_b}(A_b/M)$$. (In the process of proving this, the author uses classification of finite simple groups.)
Using this, it is shown that the reduction of $$\Gamma'$$ modulo any cofinite ideal $$I$$ of $$A_b$$ (i.e., $$| A_b/I| <\infty)$$ is $$G_{A_b}(A_b/I)$$. Thus the author obtains the main result that $$\Gamma'$$ is dense in $$G_{A_b}(\hat A_b)$$ where $$\hat A_b$$ denotes the profinite completion $$\varprojlim_{| A_b/I| <\infty}(A_b/I)$$.
Reviewer: V. Lakshmibai

##### MSC:
 14L35 Classical groups (algebro-geometric aspects) 20G15 Linear algebraic groups over arbitrary fields 17B45 Lie algebras of linear algebraic groups 14M15 Grassmannians, Schubert varieties, flag manifolds
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