# zbMATH — the first resource for mathematics

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
##### Keywords:
crystallographic groups; Bieberbach groups; holonomy groups
##### Citations:
Zbl 0381.20002; Zbl 0863.20022; Zbl 0965.20031; Zbl 0789.20058
CARAT; GAP
Full Text:
##### References:
 [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 [5] Calabi E., Bull. Amer. Math. Soc. 63 pp 135– (1957) [6] DOI: 10.1007/978-1-4613-8687-2 [7] DOI: 10.1093/qmath/37.2.177 · Zbl 0598.57014 [8] DOI: 10.1107/S010876739701547X · Zbl 1176.20051 [9] Plesken W., Experiment. Math. 9 (3) pp 407– (2000) · Zbl 0965.20031 [10] Schönert M., GAP: Groups, algorithms, and programming,, 4. ed. (1994) [11] DOI: 10.1007/BF02568759 · Zbl 0789.20058 [12] DOI: 10.1007/BF02568313 · Zbl 0863.20022 [13] DOI: 10.1007/BF02568029 · Zbl 0030.00902
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.