On \((2,3,7)\)-generation of maximal parabolic subgroups (Q2765550)

Let \(E_n(R)\) denote the elementary group over a ring \(R\) with identity. The authors prove that when \(R\) is finitely generated and \(n\), \(\widetilde n\) are large enough, the subgroup \(P_{n,\widetilde n}\) consisting of the matrices \(\left(\begin{smallmatrix} A&B\\ 0&\widetilde A\end{smallmatrix}\right)\), where \(A\in E_n(R)\), \(\widetilde A\in E_{\widetilde n}(R)\) and \(B\in\text{Mat}_{n,\widetilde n}\), is \((2,3,7)\)-generated, i.e., \(P_{n,\widetilde n}\) is an epimorphic image of the infinite triangle group \(\triangle(2,3,7)\).