Linear groups rich in finite quotients (Q1371300)

Let \({\mathbf X}\) be a class of groups. The class of counter-\({\mathbf X}\) groups is the class of all groups having no non-trivial quotients in \({\mathbf X}\). It follows that a group \(G\) is counter-counter-finite if and only if every non-trivial quotient of \(G\) has a non-trivial finite quotient. In a previous paper [Glasg. Math. J. 38, No. 3, 263-274 (1996; Zbl 0884.20019)]\ the author has proved that a group \(G\) is counter-counter-finite if and only if it is highly representable. (\(G\) is an \({\mathbf {HR}}\)-group if and only if every proper quotient of \(G\) has a nontrivial representation over any finitely generated domain.) The author determines precisely those infinite fields \(k\) for which \(\text{GL}_n(k)\) is an \({\mathbf {HR}}\)-group. He extends these results to various groups \(\text{GL}_n(R)\), where \(R\) is a commutative ring with a (finite) stable rank and \(n\) is ``suitable large''. For example he proves that \(\text{GL}_n(R)\in{\mathbf{HR}}\), when \(R\) is a finitely generated \(\mathbb{Z}\)-algebra and \(n\geq 3\).