Holonomy groups of Bieberbach groups with finite outer automorphism groups
type: Item
instance of: scholarly article
title: Holonomy groups of Bieberbach groups with finite outer automorphism groups
zbMATH Open document ID: 0834.20054
DOI: 10.1007/BF01196572
author: Gerhard Hiss
author: Andrzej Szczepański
published in: Archiv der Mathematik
publication date: 1996-03-18T00:00:00Z
review text: The paper is a first step towards the classification of holonomy groups \(G\) of Bieberbach groups \(\Gamma\) with finite outer automorphism group \(\text{Out}(\Gamma)\). In particular the authors completely classify all such abelian groups. Here a Bieberbach group of dimension \(n\) (the fundamental group of a compact connected flat manifold \(X\)) is a torsion free group \(\Gamma\) defined by a short exact sequence \(0\to L\to\Gamma\to G\to 1\), where \(G\) is finite (holonomy group of \(\Gamma\)), and \(L\) is a free maximal abelian subgroup of \(\Gamma\) of rank \(n\). In particular, the group \(\text{Aff}(X)\) of affine self equivalences of \(X\) is finite if and only if \(\text{Out}(\Gamma)\) is finite and the first Betti number of \(X\) is zero. See also \textit{H. Brown, J. Neubüser} and \textit{H. Zassenhaus} [Math. Comput. 27, 167-182 (1973; Zbl 0255.20031)].
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 779016
zbMATH Keywords: holonomy groups
zbMATH Keywords: Bieberbach groups
zbMATH Keywords: finite outer automorphism groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: compact connected flat manifolds
zbMATH Keywords: torsion free groups
zbMATH Keywords: maximal Abelian subgroups
zbMATH Keywords: affine self equivalences
zbMATH Keywords: first Betti number
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Modular representation theory: New trends and methods
cites work: Bieberbach groups and flat manifolds
cites work: Q3684278
cites work: The computations of some Schur indices
cites work: HOLONOMY OF FLAT MANIFOLDS WITH b1 = 0, II
cites work: Endliche Gruppen I
cites work: The loewy structure of the projective indecomposable modules for the mathieu groups in characteristic 3
cites work: Minimal dimensions for flat manifolds with prescribed holonomy
cites work: Anosov diffeomorphisms of flat manifolds
cites work: Localization in the classification of flat manifolds
cites work: Five dimensional Bieberbach groups with trivial centre
description: scientific article; zbMATH DE number 779016