Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups.

*(English)*Zbl 0568.14025In 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 \(\varprojlim_{| A_b/I| <\infty}(A_b/I)\).

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