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ISRN Geometry
Volume 2012 (2012), Article ID 685103, 27 pages
http://dx.doi.org/10.5402/2012/685103
Research Article

On Fundamental Domains for Subgroups of Isometries Acting in ℍ 𝑛

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), 04510 Mexico, DF, Mexico

Received 19 March 2012; Accepted 2 May 2012

Academic Editors: A. Fino and E. H. Saidi

Copyright © 2012 Antonio Lascurain Orive and Rubén Molina Hernández. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given 𝒫 a fundamental polyhedron for the action of 𝐺 , a classical kleinian group, acting in 𝑛 -dimensional hyperbolic space, and Ξ“ , a finite index subgroup of 𝐺 , one obtains a fundamental domain for Ξ“ pasting copies of 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  ℝ 𝑛 is transitive on “ π‘˜ -spheres”. Hence, describing the hyperbolic π‘˜ -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.