# Almost-Bieberbach groups with prime order holonomy

Fundamenta Mathematicae (1996)

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

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topDekimpe, 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

top- [1] A. Babakhanian, Cohomological Methods in Group Theory, Pure and Appl. Math. 11, Marcel Dekker, New York, 1972. Zbl0256.20068
- [2] K. Dekimpe, Almost Bieberbach groups: cohomology, construction and classification, Doctoral Thesis, K.U. Leuven, 1993.
- [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] 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] 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] 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] 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] W. Malfait, Symmetry of infra-nilmanifolds: an algebraic approach, Doctoral Thesis, K.U. Leuven, 1994.
- [9] D. S. Passman, The Algebraic Structure of Group Rings, Pure and Appl. Math., Wiley, New York, 1977. Zbl0368.16003
- [10] D. Segal, Polycyclic Groups, Cambridge University Press, 1983. Zbl0516.20001
- [11] A. Szczepański, Decomposition of flat manifolds, preprint, 1995.
- [12] A. T. Vasquez, Flat Riemannian manifolds, J. Differential Geom. 4 (1970), 367-382. Zbl0209.25402
- [13] S. T. Yau, Compact flat Riemannian manifolds, J. Differential Geom. 6 (1972), 395-402. Zbl0238.53025

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