Polyhedral groups and graph amalgamation products
Given a finite polyhedron P in hyperbolic 3-space \({\mathbb{H}}^ 3\), with all dihedral angles of the form \(\pi\) /n, \(n\in {\mathbb{N}}\), the reflections along the faces of P generate a properly discontinuous group \(G^*\) of hyperbolic isometries with P as fundamental region. The authors show that the subgroup G of orientation-preserving elements in \(G^*\), which they call a polyhedral group, has in a natural way the structure of a so- called graph amalgamation product: the graph is the 2-skeleton of P, the associated vertex- and edge-groups are the stabilizers in G, and amalgamation takes place over all edges of the graph; it follows that G is a factor group of the graph product of the graph of groups, in the sense of Bass-Serre, by setting all the HNN-generators equal to one. Two classes of examples are discussed: tetrahedral groups and, as the main application, the groups PSL(2,\({\mathfrak O}_ m)\) and PGL(2,\({\mathfrak O}_ m)\), where \({\mathfrak O}_ m=ring\) of integers in the imaginary quadratic number field \({\mathbb{Q}}(\sqrt{-m})\), \(m\in {\mathbb{N}}\), for various small values of m (''Bianchi groups''). Many of these are polyhedral groups or are closely related to polyhedral groups. The paper contains many interesting structural properties about these groups and certain subgroups. It is carefully written and pleasant to read. A related paper by the authors and \textit{M. Frame} studies torsion-free subgroups of the Picard group PSL(2,\({\mathfrak O}_ 1)\), using the graph amalgamation structure of this group [Trans. Am. Math. Soc. 282, 205-235 (1984; Zbl 0544.57005)].
properly discontinuous group of hyperbolic isometries
torsion-free subgroups of the Picard group
scientific article; zbMATH DE number 3911325
scientific article