scholarly journals Compact flat manifolds with holonomy group $\mathbf {Z}_2 \oplus \mathbf {Z}_2$

1996 ◽  
Vol 124 (8) ◽  
pp. 2491-2499 ◽  
Author(s):  
J. P. Rossetti ◽  
P. A. Tirao
Keyword(s):  
Holonomy Group ◽  
Compact Flat Manifolds ◽  
Flat Manifolds

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.

scholarly journals Length spectra andp-spectra of compact flat manifolds

2003 ◽  
Vol 13 (4) ◽  
pp. 631-657 ◽  
Author(s):  
R. J. Miatello ◽  
J. P. Rossetti
Keyword(s):  
Compact Flat Manifolds ◽  
Flat Manifolds

Isospectral compact flat manifolds

1992 ◽  
Vol 68 (3) ◽  
pp. 489-498 ◽  
Author(s):  
Isabel Dotti Miatello ◽  
Roberto J. Miatello
Keyword(s):  
Compact Flat Manifolds ◽  
Flat Manifolds

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 Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 357-363
Author(s):  
Emilio A. Lauret
Keyword(s):  
Isometry Group ◽  
Regular Representations ◽  
Compact Flat Manifolds ◽  
The Right ◽  
Flat Manifolds ◽  
Full Isometry Group

AbstractLet Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of ℝn. We prove that if the compact flat manifolds Γ1\ℝn and Γ2\ℝn are strongly isospectral, then the Bieberbach groups Γ1 and Γ2 are representation equivalent; that is, the right regular representations L2(Γ1\G) and L2(Γ2\G) are unitarily equivalent.

scholarly journals FINITE FLAT SPACES

Mathematika ◽  
2019 ◽  
Vol 65 (4) ◽  
pp. 1010-1017
Author(s):  
Vladimir Zolotov
Keyword(s):  
Lie Groups ◽  
Metric Spaces ◽  
List Type ◽  
Euclidean Spaces ◽  
Markov Type ◽  
Finite Metric Space ◽  
Isometric Actions ◽  
Compact Flat Manifolds ◽  
Flat Manifolds

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.(a)Quotients of Euclidean spaces by isometric actions of finite groups.(b)$L_{2}$-Wasserstein spaces over Euclidean spaces.(c)Compact flat manifolds.(d)Compact flat orbifolds.(e)Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

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