Finitely generated profinitely dense free groups in higher rank semi-simple groups
Let \(G\) be a non-compact linear semi-simple group and \(\Gamma\) an arithmetic subgroup. If the associated simply connected algebraic group over \({\mathbb Q}\) for \(\Gamma\) has the congruence subgroup property, the authors prove \(\Gamma\) contains a finitely generated profinitely dense free (\textit{fgpdf}) subgroup. Here profinitely dense means that the subgroup maps onto every finite quotient of \(\Gamma\). The congruence subgroup property is defined in the paper, but holds for most groups not of type \(A_n\). This work answers a question posed by Shalev regarding previous work of Margulis and Soifer. It also shows the existence of \textit{fgpdf} subgroup of \(SL_n({\mathbb Z})\) for \(n\geq 3\). More generally, they prove that \(SL_n({\mathbb Z})\) may be replaced by any irreducible arithmetic non-compact lattice in a higher rank group.
irreducible arithmetic non-compact lattice
scientific article; zbMATH DE number 1435999
scientific article