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Computation of five- and six-dimensional Bieberbach groups. (English) Zbl 0980.20043
An \(n\)-dimensional crystallographic group is a discrete, cocompact subgroup of \(\text{Isom}(\mathbb{R}^n)\). Torsion-free crystallographic groups are called Bieberbach groups. From the Bieberbach theorem there is only a finite number of such groups of given dimension up to isomorphism. Moreover, any such group has a maximal Abelian subgroup of finite index. We call the finite quotient group the holonomy group of the crystallographic group. The full classification is known for \(n\leq 4\) [cf. H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic groups of four-dimensional space, Wiley (1978; Zbl 0381.20002)]. For \(n=5\) it was known the list of holonomy groups [cf. A. Szczepański, Manuscr. Math. 90, No. 3, 383-389 (1996; Zbl 0863.20022)] in the torion-free case (Bieberbach groups).
In the paper the classification of all Bieberbach groups of dimension 5 and 6 is given. The authors have written some computer program which produces the above list of groups. The program is an application of extension methods from H. Brown et al. [loc. cit.]; J. Opgenorth, W. Plesken and T. Schulz [Crystallographic algorithms and tables, Acta Cryst., Sect. A 54, No. 5, 517-531 (1998)]; W. Plesken and T. Schulz [Exp. Math. 9, No. 3, 407-411 (2000; Zbl 0965.20031)] which all come from Zassenhaus’s algorithm.
In the paper are only the final tables, the full lists can be found at CARAT [cf. J. Opgenorth et al. (loc. cit.)]. Also important seems to be a list of groups with trivial center which has its roots in the Calabi methods of the classification of flat manifolds [cf. A. Szczepański, Manuscr. Math. 68, No. 2, 191-208 (1990; Zbl 0789.20058)]. Most calculations are made on a computer.

20H15 Other geometric groups, including crystallographic groups
20-04 Software, source code, etc. for problems pertaining to group theory
57M07 Topological methods in group theory
[1] DOI: 10.2307/1970053 · Zbl 0079.38304
[2] DOI: 10.1007/BF01564500 · JFM 42.0144.02
[3] DOI: 10.1007/BF01456724 · JFM 43.0186.01
[4] Brown H., Crystallographic groups of four-dimensional space (1978) · Zbl 0381.20002
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[8] DOI: 10.1107/S010876739701547X · Zbl 1176.20051
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[11] DOI: 10.1007/BF02568759 · Zbl 0789.20058
[12] DOI: 10.1007/BF02568313 · Zbl 0863.20022
[13] DOI: 10.1007/BF02568029 · Zbl 0030.00902
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