Hyperbolic 3-Manifolds which share a Fundamental Polyhedron

1981 ◽  
pp. 505-514 ◽  
Author(s):  
Norbert J. Wielenberg
Keyword(s):  
Fundamental Polyhedron

The Role of VLBI in the Establishment of Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 293-295 ◽  
Author(s):  
I. Zhongolovitch
Keyword(s):  
General Solution ◽  
Future Development ◽  
Rational Approach ◽  
Radio Telescopes ◽  
Coordinate Systems ◽  
Tectonic Plates ◽  
Astronomical Observations ◽  
Optical Instruments ◽  
Fundamental Polyhedron

Considering the future development and general solution of the problem under consideration and also the high precision attainable by astronomical observations, the following procedure may be the most rational approach:1. On the main tectonic plates of the Earth’s crust, powerful movable radio telescopes should be mounted at the same points where standard optical instruments are installed. There should be two stations separated by a distance of about 6 to 8000 kilometers on each plate. Thus, we obtain a fundamental polyhedron embracing the whole Earth with about 10 to 12 apexes, and with its sides represented by VLBI.

Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension

1996 ◽  
Vol 120 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Sanju L. Velani
Keyword(s):  
Hyperbolic Space ◽  
Finite Groups ◽  
Discrete Subgroup ◽  
Dimensional Euclidean Space ◽  
Simultaneous Diophantine Approximation ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Fundamental Polyhedron ◽  
Image Position ◽  
Geometrically Finite Groups

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

scholarly journals On Fundamental Domains for Subgroups of Isometries Acting in

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-27
Author(s):  
Antonio Lascurain Orive ◽  
Rubén Molina Hernández
Keyword(s):  
Alternative Proof ◽  
Index Subgroup ◽  
Transitive Action ◽  
Space Model ◽  
Fundamental Domains ◽  
Möbius Group ◽  
Fundamental Polyhedron ◽  
Hyperbolic Planes ◽  
Generating Theorem ◽  
Finite Index Subgroup

Given a fundamental polyhedron for the action of , a classical kleinian group, acting in -dimensional hyperbolic space, and , a finite index subgroup of , one obtains a fundamental domain for 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 is transitive on “-spheres”. Hence, describing the hyperbolic -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.

scholarly journals On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4

2019 ◽  
Vol 486 (1) ◽  
pp. 7-11
Author(s):  
N. V. Bogachev
Keyword(s):  
Three Dimensional ◽  
Reflection Group ◽  
Lobachevsky Space ◽  
Fundamental Polyhedron

In this paper we prove that the fundamental polyhedron of a ℤ2-arithmetic reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of stably reflective hyperbolic ℤ2-lattices of rank 4.

scholarly journals CoxIter – Computing invariants of hyperbolic Coxeter groups

2015 ◽  
Vol 18 (1) ◽  
pp. 754-773 ◽  
Author(s):  
R. Guglielmetti
Keyword(s):  
Computer Program ◽  
Euler Characteristic ◽  
Coxeter Groups ◽  
Source Code ◽  
Theoretical Background ◽  
Combinatorial Structure ◽  
Fundamental Polyhedron ◽  
Supplementary Material

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).Supplementary materials are available with this article.

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