The finite subgroups of maximal arithmetic Kleinian groups
The finite subgroups of \(\text{PGL}(2,\mathbb{C})\) have been known since Klein's time to be isomorphic to a cyclic group, a dihedral group, \(A_4\), \(S_4\) or \(A_5\). It is useful to know which of these finite groups appear as subgroups of a given Kleinian group \(\Gamma\subset\text{PGL}(2,\mathbb{C})\). The authors treat here the more restricted problem of computing the finite subgroups of a Kleinian group corresponding to a minimal arithmetic hyperbolic 3-orbifold, where minimal means that the orbifold does not properly cover any other orbifold. This subject is linked to interesting aspects of indefinite quaternion algebras over a number field and of an assignment, linked to dihedral subgroups of arithmetic Kleinian groups, of an ideal class to a pair of global Hilbert symbols. They give a lot of interesting primers. Note that while their main interest is in finite subgroups of a maximal arithmetic Kleinian group \(\Gamma\subset\text{PGL}(2,\mathbb{C})\), the authors actually deal throughout with the more general case of a maximal irreducible arithmetic group \(\Gamma\subset\text{PGL}(2,\mathbb{R})^n\times\text{PGL}(2,\mathbb{C})^m\).
minimal arithmetic hyperbolic 3-orbifolds
scientific article; zbMATH DE number 1544081
scientific article