scholarly journals On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane

2005 ◽  
Vol 187 (2) ◽  
pp. 95-110 ◽  
Author(s):  
G. R. Conner ◽  
J. W. Lamoreaux
Keyword(s):  
Euclidean Plane ◽  
Metric Spaces ◽  
Universal Covering ◽  
Covering Spaces

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.

scholarly journals Generalized universal covering spaces and the shape group

2007 ◽  
Vol 197 ◽  
pp. 167-196 ◽  
Author(s):  
Hanspeter Fischer ◽  
Andreas Zastrow
Keyword(s):  
Universal Covering ◽  
Covering Spaces ◽  
Shape Group

scholarly journals A note on the variation of Riemann surfaces

1996 ◽  
Vol 142 ◽  
pp. 1-4 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Riemann Surface ◽  
Complex Plane ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Universal Covering ◽  
Covering Space ◽  
Uniformization Theorem ◽  
Covering Spaces ◽  
Universal Covering Space

Let X be any Riemann surface. By Koebe’s uniformization theorem we know that the universal covering space of X is conformally equivalent to either Riemann sphere, complex plane, or the unit disc in the complex plane. If X is allowed to vary with parameters we may inquire the parameter dependence of the corresponding family of the universal covering spaces.

scholarly journals A Remark on Differentiable Structures on Real Projective (2n-1)-Spaces

1966 ◽  
Vol 27 (1) ◽  
pp. 357-360 ◽  
Author(s):  
Kenichi Shiraiwa
Keyword(s):  
Universal Covering ◽  
Connected Sum ◽  
Covering Spaces ◽  
The Real

The main objective of this paper is to study the action of the group of differentiate structures Γ2n-1 on the (2n-1)-sphere S2n-1 on the diffeomorphism classes on the real projective (2n-1)-space P2n-1 by connected sum. This is done by considering universal covering spaces of the connected sum where Σ is an exotic (2n-1)-sphere.

scholarly journals Hyperbolic distance versus quasihyperbolic distance in plane domains

2021 ◽  
Vol 8 (20) ◽  
pp. 578-614
Author(s):  
David Herron ◽  
Jeff Lindquist
Keyword(s):  
Euclidean Plane ◽  
Metric Spaces ◽  
The Other ◽  
Hyperbolic Distance ◽  
Other Hand ◽  
Gromov Hyperbolic ◽  
Hyperbolic Domain ◽  
Hyperbolic Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

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