On Fundamental Domains for Subgroups of Isometries Acting in ℍn

Abstract EN

Given ? a fundamental polyhedron for the action of G, a classical kleinian group, acting in n-dimensional hyperbolic space, and Γ, a finite index subgroup of G, one obtains a fundamental domain for Γ pasting copies of ? 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 ℝ^n is transitive on “k-spheres”.

Hence, describing the hyperbolic k-planes in the upper half space model intrinsically, and providing 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.