Computational aspects of group extensions and their applications in topology. (English) Zbl 1101.20302

Summary: We describe algorithms to determine extensions of infinite polycyclic groups having certain properties. In particular, we are interested in torsion-free extensions and extensions whose Fitting subgroup has a minimal centre. Then we apply these methods in topological applications. We discuss the calculation of Betti numbers for compact manifolds, and we describe algorithmic approaches in classifying almost Bieberbach groups.


20F16 Solvable groups, supersolvable groups
20-04 Software, source code, etc. for problems pertaining to group theory
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
20H15 Other geometric groups, including crystallographic groups
68W30 Symbolic computation and algebraic computation


Polycyclic; AClib; GAP
Full Text: DOI EuDML


[1] Brown H., Crystallographic groups of four-dimensional Space. (1978) · Zbl 0381.20002
[2] Brown K. S., Cohomology of groups, volume 87 of Grad. Texts in Math. (1982) · Zbl 0584.20036
[3] Conner, P. E. and Raymond, F. 1971.”Actions of compact Lie groups on aspherical manifolds.”227–264. [Conner and Raymond 71], inTopoloqy of Manifolds, Proc. Univ. Georqia 1969
[4] DOI: 10.1090/S0002-9904-1977-14179-7 · Zbl 0341.57003
[5] Dekimpe K., Almost–Bieberbach Groups: Affine and Polynomial Structures (1996) · Zbl 0865.20001
[6] Dekimpe K., Aclib – A GAP 4 package (2001)
[7] Dekimpe, K. and Eick, B. ”Computations with almost-crystallographic groups.”. Proceedings of Groups. 2001, St. Andrews. Edited by: Campbell and Robertson. [Dekimpe and Eick 01b] · Zbl 1054.20031
[8] Dekimpe K., Invent. Math. 129 (1) pp 121– (1997) · Zbl 0867.20031
[9] Dekimpe K., Quart. J. Math. 46 (2) pp 141– (1995) · Zbl 0854.57014
[10] Eick, B. ”Computing with infinite polycyclic groups.”. Proceedings Groups and Computation III. Edited by: Seress and Kantor. pp.139–153. Berlin: de Gruyter. [Eick Ola]
[11] Eick B., Algorithms for polycyclic groups. (2001) · Zbl 0991.20028
[12] Eick B., Polycyclic – A Gap 4 package (2000)
[13] Algorithms and Programming, Version 4.2 (2000)
[14] Goze M., Mathematics and Its Applications (1996)
[15] Hahn T., International tables for crystallography, A. Space-group symmetry, (International union of crystallography) (1987) · Zbl 1371.82119
[16] DOI: 10.1093/qmath/39.1.61 · Zbl 0655.57029
[17] Lee K. B., ”Infranil-manifolds modeled on Heis5,” (2000)
[18] Lee K. B, Geom. Dedicata 87 (1) pp 167– (2000)
[19] Robinson D. J. S., A course in group theory. (1982)
[20] DOI: 10.1017/CBO9780511565953
[21] DOI: 10.1017/CBO9780511574702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.