The finite subgroups of maximal arithmetic Kleinian groups. (English) Zbl 0973.20040

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 \(\Gamma\subset\text{PGL}(2,\mathbb{R})^n\times\text{PGL}(2,\mathbb{C})^m\).


20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11R52 Quaternion and other division algebras: arithmetic, zeta functions
11F06 Structure of modular groups and generalizations; arithmetic groups
20E07 Subgroup theorems; subgroup growth
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