The Bianchi groups are separable on geometrically finite subgroups. (Q1826269)

Let \(G\) be a group and \(H\) a finitely generated subgroup; \(G\) is called `\(H\)-subgroup separable' if for every \(g\in G\setminus H\) there is a subgroup \(K\subset G\) of finite index such that \(H\subset K\) and \(g\notin K\). A Kleinian group \(G\) is called `geometrically finite' if \(G\) admits a finite-sided Dirichlet polyhedron for its action on \(\mathbb{H}^3\). The main result is that the group \(\text{PSL}(2,{\mathcal O}_d)\) is \(H\)-subgroup separable for geometrically finite subgroups \(H\). Here \(d\) is any square free positive integer and \({\mathcal O}_d\) is the ring of integers in \(\mathbb{Q}(\sqrt{-d})\).