Almost-Bieberbach groups with prime order holonomy

• Autorzy: Karel Dekimpe , Wim Malfait
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 20F34: Fundamental groups and their automorphisms
• 20F17: Formations of groups, Fitting classes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 151
• Numer: 2
• Strony: 167-176

Daty

wydano

1996

otrzymano

1995-11-21

poprawiono

1996-06-26

Twórcy

autor

Karel Dekimpe

• Katholieke Universiteit Leuven, Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium

autor

Wim Malfait

• Katholieke Universiteit Leuven, Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium

Bibliografia

• [1] A. Babakhanian, Cohomological Methods in Group Theory, Pure and Appl. Math. 11, Marcel Dekker, New York, 1972.
• [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.
• [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.
• [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.
• [6] K. B. Lee, There are only finitely many infra-nilmanifolds under each nilmanifold, Quart. J. Math. Oxford (2) 39 (1988), 61-66.
• [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.
• [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.
• [10] D. Segal, Polycyclic Groups, Cambridge University Press, 1983.
• [11] A. Szczepański, Decomposition of flat manifolds, preprint, 1995.
• [12] A. T. Vasquez, Flat Riemannian manifolds, J. Differential Geom. 4 (1970), 367-382.
• [13] S. T. Yau, Compact flat Riemannian manifolds, J. Differential Geom. 6 (1972), 395-402.