scholarly journals Resilient Routing for Sensor Networks Using Hyperbolic Embedding of Universal Covering Space

2010 ◽  
Author(s):  
Wei Zeng ◽  
Rik Sarkar ◽  
Feng Luo ◽  
Xianfeng Gu ◽  
Jie Gao
Keyword(s):  
Sensor Networks ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space ◽  
Hyperbolic Embedding

A quadratic parabolic group

1975 ◽  
Vol 77 (2) ◽  
pp. 281-288 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Normal Form ◽  
Half Space ◽  
Transformation Group ◽  
Universal Covering ◽  
Covering Space ◽  
Fundamental Domains ◽  
Universal Covering Space ◽  
Other Information ◽  
Figure Eight Knot

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

Rigidities of asymptotically Euclidean manifolds

1999 ◽  
Vol 60 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Group ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

scholarly journals A presentation for a group of automorphisms of a simplicial complex

1988 ◽  
Vol 30 (3) ◽  
pp. 331-337 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Simplicial Complex ◽  
Elementary Proof ◽  
Universal Covering ◽  
Covering Space ◽  
Covering Spaces ◽  
Group Of Automorphisms ◽  
Generators And Relations ◽  
Universal Covering Space ◽  
Simply Connected ◽  
Special Case

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

On Wh3 of a Bieberbach group

1984 ◽  
Vol 95 (1) ◽  
pp. 55-60
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Universal Covering ◽  
Closed Manifold ◽  
Covering Space ◽  
Whitehead Group ◽  
Bieberbach Group ◽  
Universal Covering Space ◽  
Aspherical Manifold

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

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