scholarly journals Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

2021 ◽  
Author(s):  
Oscar Ocampo
Keyword(s):  
Fundamental Group ◽  
Abelian Subgroup ◽  
Holonomy Group ◽  
Braid Groups ◽  
Crystallographic Group ◽  
Flat Manifold ◽  
Kahler Geometry ◽  
Bieberbach Group ◽  
Odd Order ◽  
Flat Manifolds

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

scholarly journals Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Genildo de Jesus Nery
Keyword(s):  
Fundamental Group ◽  
Prime Order ◽  
Isomorphism Class ◽  
Holonomy Group ◽  
Fundamental Groups ◽  
Flat Manifold ◽  
Profinite Completion ◽  
Compact Flat Manifolds ◽  
Flat Manifolds

Abstract In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

On the Abelianization of a Torsion Free Crystallographic Group

2014 ◽  
Vol 70 (1) ◽  
Author(s):  
Nor'ashiqin Mohd Idrus ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim ◽  
Rohaidah Masri
Keyword(s):  
Fundamental Group ◽  
Riemannian Manifolds ◽  
Point Group ◽  
Crystallographic Group ◽  
Finite Point ◽  
Lattice Group ◽  
Bieberbach Group ◽  
Trivial Center ◽  
Free Abelian Groups ◽  
Torsion Free

A torsion free crystallographic group, which is also known as a Bieberbach group is a generalization of free abelian groups. It is an extension of a lattice group by a finite point group. The study of n-dimensional crystallographic group had been done by many researchers over a hundred years ago. A Bieberbach group has been characterized as a fundamental group of compact, connected, flat Riemannian manifolds. In this paper, we characterize Bieberbach groups with trivial center as exactly those with finite abelianizations.  The abelianization of a Bieberbach group is shown to be finite if the center of the group is trivial.

scholarly journals Almost-Bieberbach groups with (in)finite outer automorphism group

1998 ◽  
Vol 40 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Wim Malfait ◽  
Andrzej Szczepański
Keyword(s):  
Automorphism Group ◽  
Fundamental Group ◽  
Infinite Order ◽  
Outer Automorphism ◽  
Crystallographic Group ◽  
Outer Automorphism Group ◽  
Outer Automorphisms ◽  
Bieberbach Group ◽  
Interesting Class ◽  
Almost Bieberbach Group

AbstractIf we investigate symmetry of an infra-nilmanifoldM, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groupsEwith finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms onM, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group ofE.

scholarly journals Crystallographic groups and flat manifolds from surface braid groups

2020 ◽  
pp. 107560
Author(s):  
Daciberg Lima Gonçalves ◽  
John Guaschi ◽  
Oscar Ocampo ◽  
Carolina de Miranda e Pereiro
Keyword(s):  
Braid Groups ◽  
Crystallographic Groups ◽  
Flat Manifolds

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

scholarly journals An almost flat manifold with a cyclic or quaternionic holonomy group bounds

2016 ◽  
Vol 103 (2) ◽  
pp. 289-296 ◽  
Author(s):  
James F. Davis ◽  
Fuquan Fang
Keyword(s):  
Holonomy Group ◽  
Flat Manifold

scholarly journals Käahler identity on Levi flat manifolds and application to the embedding

2000 ◽  
Vol 158 ◽  
pp. 87-93 ◽  
Author(s):  
Takeo Ohsawa ◽  
Nessim Sibony
Keyword(s):  
Line Bundle ◽  
Finite Order ◽  
Flat Manifold ◽  
Positive Line Bundle ◽  
Flat Manifolds ◽  
Positive Line

AbstractIt is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into ℙn by a CR mapping with an arbitrarily high, though finite, order of regularity.

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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