Link complements and the Bianchi modular groups (Q2716137)

Among the most natural examples of discrete groups of isometries of hyperbolic \(3\)-space \(\mathbb{H}^3\) are the Bianchi groups. These are the groups \(\text{PSL}(2,{\mathcal O}_m)\), where \({\mathcal O}_m\) is the ring of integers in the quadratic number field \(\mathbb{Q}\sqrt{-m}\). Any torsion-free subgroup \(\Gamma\) of finite index in \(\text{PSL}(2,{\mathcal O}_m)\) is the fundamental group of a complete hyperbolic \(3\)-manifold of finite volume. Cohomological results of \textit{K. Vogtmann} [Math. Ann. 272, 399-419 (1985; Zbl 0553.20023)] show that these manifolds cannot be the complements of links in \(S^3\) unless \(m\) is in \(\{1,2,3,5,6,7,11,15,19,23,31,39,47,71\}\). Various authors had shown that link complements can indeed arise for the values of \(m\) in \(\{1,2,3,7,11,15,23\}\). In this paper, examples of such link complements are obtained for the remaining possible values of \(m\) as well.NEWLINENEWLINENEWLINEFor each of the remaining values of \(m\), the underlying topological space of the quotient orbifold \(\Omega_m=\mathbb{H}^3/\text{PSL}(2,{\mathcal O}_m)\) imbeds as the complement of a link \(L_m\) into \(S^3\). Using \textit{R. Riley}'s computer-assisted construction of the Ford domains for the Bianchi groups [Math. Comput. 40, 607-632 (1983; Zbl 0528.51010)] and similar imbeddings of the closely related orbifolds \(\mathbb{H}^3/\text{PGL}(2,{\mathcal O}_m)\) due to A. Hatcher, explicit pictures are obtained for these links together with the singular sets of the orbifolds. The author then forms certain branched coverings of \(S^3\) over \(S^3\), branched over the singular sets of the \(\Omega_m\). The complement of the preimage \(\widetilde L_m\) of \(L_m\) is the desired hyperbolic manifold. Explicit pictures of these links, many of them rather complicated, are given for most cases. The volumes were checked using J. Weeks' SnapPea program and found to be consistent with the volumes already known for the Bianchi orbifolds.