Almost-Bieberbach groups with prime order holonomy

Karel Dekimpe; Wim Malfait

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 2, page 167-176
  • ISSN: 0016-2736

Abstract

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The main issue of this paper is an attempt to find a decomposition theorem for infra-nilmanifolds in the same spirit as a result of A. Vasquez for flat Riemannian manifolds. That is: we look for infra-nilmanifolds with prime order holonomy which can be obtained as a fiber space with a non-trivial nilmanifold as fiber and an infra-nilmanifold as its base.  In this perspective, we prove the following algebraic result: if E is an almost-Bieberbach group with prime order holonomy, then there is a normal subgroup Π of E contained in the Fitting subgroup of E such that E/Π is an almost-Bieberbach group either having a Fitting subgroup with center isomorphic to the infinite cyclic group, or having an underlying crystallographic group with torsion and a center coinciding with that of its Fitting subgroup.

How to cite

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Dekimpe, Karel, and Malfait, Wim. "Almost-Bieberbach groups with prime order holonomy." Fundamenta Mathematicae 151.2 (1996): 167-176. <http://eudml.org/doc/212188>.

@article{Dekimpe1996,
abstract = { The main issue of this paper is an attempt to find a decomposition theorem for infra-nilmanifolds in the same spirit as a result of A. Vasquez for flat Riemannian manifolds. That is: we look for infra-nilmanifolds with prime order holonomy which can be obtained as a fiber space with a non-trivial nilmanifold as fiber and an infra-nilmanifold as its base.  In this perspective, we prove the following algebraic result: if E is an almost-Bieberbach group with prime order holonomy, then there is a normal subgroup Π of E contained in the Fitting subgroup of E such that E/Π is an almost-Bieberbach group either having a Fitting subgroup with center isomorphic to the infinite cyclic group, or having an underlying crystallographic group with torsion and a center coinciding with that of its Fitting subgroup. },
author = {Dekimpe, Karel, Malfait, Wim},
journal = {Fundamenta Mathematicae},
keywords = {almost-Bieberbach groups; infra-nilmanifolds; fiber spaces; Fitting subgroup; crystallographic groups},
language = {eng},
number = {2},
pages = {167-176},
title = {Almost-Bieberbach groups with prime order holonomy},
url = {http://eudml.org/doc/212188},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Dekimpe, Karel
AU - Malfait, Wim
TI - Almost-Bieberbach groups with prime order holonomy
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 2
SP - 167
EP - 176
AB - The main issue of this paper is an attempt to find a decomposition theorem for infra-nilmanifolds in the same spirit as a result of A. Vasquez for flat Riemannian manifolds. That is: we look for infra-nilmanifolds with prime order holonomy which can be obtained as a fiber space with a non-trivial nilmanifold as fiber and an infra-nilmanifold as its base.  In this perspective, we prove the following algebraic result: if E is an almost-Bieberbach group with prime order holonomy, then there is a normal subgroup Π of E contained in the Fitting subgroup of E such that E/Π is an almost-Bieberbach group either having a Fitting subgroup with center isomorphic to the infinite cyclic group, or having an underlying crystallographic group with torsion and a center coinciding with that of its Fitting subgroup.
LA - eng
KW - almost-Bieberbach groups; infra-nilmanifolds; fiber spaces; Fitting subgroup; crystallographic groups
UR - http://eudml.org/doc/212188
ER -

References

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  1. [1] A. Babakhanian, Cohomological Methods in Group Theory, Pure and Appl. Math. 11, Marcel Dekker, New York, 1972. Zbl0256.20068
  2. [2] K. Dekimpe, Almost Bieberbach groups: cohomology, construction and classification, Doctoral Thesis, K.U. Leuven, 1993. 
  3. [3] K. Dekimpe, P. Igodt, S. Kim and K. B. Lee, Affine structures for closed 3-dimensional manifolds with NIL-geometry, Quart. J. Math. Oxford (2) 46 (1995), 141-167. Zbl0854.57014
  4. [4] K. Dekimpe, P. Igodt and W. Malfait, On the Fitting subgroup of almost crystallographic groups, Tijdschrift van het Belgisch Wiskundig Genootschap, 1993, B 1, 35-47. Zbl0802.20040
  5. [5] Y. Kamishima, K. B. Lee and F. Raymond, The Seifert construction and its applications to infra-nilmanifolds, Quart. J. Math. Oxford (2) 34 (1983), 433-452. Zbl0542.57013
  6. [6] K. B. Lee, There are only finitely many infra-nilmanifolds under each nilmanifold, Quart. J. Math. Oxford (2) 39 (1988), 61-66. Zbl0655.57029
  7. [7] K. B. Lee and F. Raymond, Geometric realization of group extensions by the Seifert construction, in: Contemp. Math. 33, Amer. Math. Soc., 1984, 353-411. Zbl0554.57021
  8. [8] W. Malfait, Symmetry of infra-nilmanifolds: an algebraic approach, Doctoral Thesis, K.U. Leuven, 1994. 
  9. [9] D. S. Passman, The Algebraic Structure of Group Rings, Pure and Appl. Math., Wiley, New York, 1977. Zbl0368.16003
  10. [10] D. Segal, Polycyclic Groups, Cambridge University Press, 1983. Zbl0516.20001
  11. [11] A. Szczepański, Decomposition of flat manifolds, preprint, 1995. 
  12. [12] A. T. Vasquez, Flat Riemannian manifolds, J. Differential Geom. 4 (1970), 367-382. Zbl0209.25402
  13. [13] S. T. Yau, Compact flat Riemannian manifolds, J. Differential Geom. 6 (1972), 395-402. Zbl0238.53025

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