On Fundamental Domains for Subgroups of Isometries Acting in

Given a fundamental polyhedron for the action of , a classical kleinian group, acting in -dimensional hyperbolic space, and , a finite indexsubgroup of , one obtains a fundamental domain for pasting copies of by a Schreier process. It also generalizes the side pairinggenerating theorem for exact or inexact polyhedra. It is provedas well that the general Möbius group acting in is transitive on “-spheres”. Hence, describing the hyperbolic -planes in the upper half spacemodel intrinsically, and providing also an alternative proof of the transitiveaction on them. Some examples are given indetail, derived from the classical modular group and the Picard group.