Centralizers of reflections in crystallographic groups

1982 ◽  
Vol 92 (1) ◽  
pp. 79-91 ◽  
Author(s):  
T. Baskan ◽  
A. M. Macbeath
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Plane ◽  
Universal Covering ◽  
Interesting Case ◽  
Discrete Groups ◽  
Incompressible Surface ◽  
Crystallographic Groups ◽  
Incompressible Surfaces ◽  
Universal Covering Space ◽  
Manifold Theory

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic 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.

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.

The Classification of Non-Euclidean Plane Crystallographic Groups

1967 ◽  
Vol 19 ◽  
pp. 1192-1205 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Quotient Space ◽  
Fuchsian Groups ◽  
Discrete Groups ◽  
Crystallographic Groups ◽  
Algebraic Classification ◽  
Compact Quotient

This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).

An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups

1986 ◽  
Vol 99 (2) ◽  
pp. 239-246 ◽  
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Infinite Family ◽  
Universal Covering ◽  
Covering Space ◽  
Homotopy Equivalence ◽  
Universal Covering Space ◽  
Aspherical Manifold ◽  
Aspherical Manifolds ◽  
Whitehead Groups

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,and its analogue in algebraic K-theory:Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

scholarly journals On 4-manifolds with universal covering space a compact geometric manifold

1993 ◽  
Vol 55 (2) ◽  
pp. 137-148
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Homotopy Type ◽  
Universal Covering ◽  
Total Space ◽  
Covering Space ◽  
Compact Type ◽  
Universal Covering Space

AbstractThere are 11 closed 4-manifolds which admit geometries of compact type (S4, CP2 or S2 × S2) and two other closely related bundle spaces (S2 × S2 and the total space of the nontrivial RP2-bundle over S2). We show that the homotopy type of such a manifold is determined up to an ambiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.

Spherical Mean and the Fundamental Group

1991 ◽  
Vol 34 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Random Walks ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Spherical Means ◽  
Universal Covering Space ◽  
Spherical Mean

AbstractWe investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

scholarly journals On macroscopic dimension of non-spin 4-manifolds

2020 ◽  
pp. 1-10
Author(s):  
Michelle Daher ◽  
Alexander Dranishnikov
Keyword(s):  
Fundamental Group ◽  
Universal Covering ◽  
Residually Finite ◽  
Finite Fundamental Group

We prove that for 4-manifolds [Formula: see text] with residually finite fundamental group and non-spin universal covering [Formula: see text], the inequality [Formula: see text] implies the inequality [Formula: see text]. This allows us to complete the proof of Gromov’s Conjecture for 4-manifolds with abelian fundamental group.

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.

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