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.
@article{bwmeta1.element.bwnjournal-article-fmv151i2p167bwm,
author = {Karel Dekimpe and Wim Malfait},
title = {Almost-Bieberbach groups with prime order holonomy},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {167-176},
zbl = {0874.20034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p167bwm}
}
Dekimpe, Karel; Malfait, Wim. Almost-Bieberbach groups with prime order holonomy. Fundamenta Mathematicae, Tome 149 (1996) pp. 167-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p167bwm/
[00000] [1] A. Babakhanian, Cohomological Methods in Group Theory, Pure and Appl. Math. 11, Marcel Dekker, New York, 1972. | Zbl 0256.20068
[00001] [2] K. Dekimpe, Almost Bieberbach groups: cohomology, construction and classification, Doctoral Thesis, K.U. Leuven, 1993.
[00002] [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. | Zbl 0854.57014
[00003] [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. | Zbl 0802.20040
[00004] [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. | Zbl 0542.57013
[00005] [6] K. B. Lee, There are only finitely many infra-nilmanifolds under each nilmanifold, Quart. J. Math. Oxford (2) 39 (1988), 61-66. | Zbl 0655.57029
[00006] [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. | Zbl 0554.57021
[00007] [8] W. Malfait, Symmetry of infra-nilmanifolds: an algebraic approach, Doctoral Thesis, K.U. Leuven, 1994.
[00008] [9] D. S. Passman, The Algebraic Structure of Group Rings, Pure and Appl. Math., Wiley, New York, 1977. | Zbl 0368.16003
[00009] [10] D. Segal, Polycyclic Groups, Cambridge University Press, 1983. | Zbl 0516.20001
[00010] [11] A. Szczepański, Decomposition of flat manifolds, preprint, 1995.
[00011] [12] A. T. Vasquez, Flat Riemannian manifolds, J. Differential Geom. 4 (1970), 367-382. | Zbl 0209.25402
[00012] [13] S. T. Yau, Compact flat Riemannian manifolds, J. Differential Geom. 6 (1972), 395-402. | Zbl 0238.53025