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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.

MSC:
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
Software:
CARAT; GAP
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References:
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