On fundamental domains for subgroups of isometries acting in \(\mathbb H^n\)
Summary: Given \(\mathcal P\), a fundamental polyhedron for the action of \(G\), a classical Kleinian group, acting in \(n\)-dimensional hyperbolic space, and \(\Gamma\), a finite index subgroup of \(G\), one obtains a fundamental domain for \(\Gamma\) pasting copies of \(\mathcal P\) by a Schreier process. It also generalizes the side pairing generating theorem for exact or inexact polyhedra. It is proved as well that the general Möbius group acting in \(\widehat{\mathbb R}^n\) is transitive on ``\(k\)-spheres''. Hence, we describe the hyperbolic \(k\)-planes in the upper half space model intrinsically, and provide also an alternative proof of the transitive action on them. Some examples are given in detail, derived from the classical modular group and the Picard group.
scientific article; zbMATH DE number 6078640
scientific article