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On Bieberbach's analysis of discrete Euclidean groups


Author: R. K. Oliver
Journal: Proc. Amer. Math. Soc. 80 (1980), 15-21
MSC: Primary 20H15; Secondary 22E40, 51M20
DOI: https://doi.org/10.1090/S0002-9939-1980-0574501-7
MathSciNet review: 574501
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Abstract: For a subgroup G of the euclidean group $ {E_n} = {O_n} \cdot {{\mathbf{R}}^n}$ (semidirect product) and a real number $ r > 0$, let $ {G^ \ast }$ denote the translation subgroup of G, $ {G_r}$ the group generated by all (A, a) in G with $ \left\Vert {1 - A} \right\Vert < r$ (operator norm), and $ {k_n}(r)$ the maximum number of elements of $ {O_n}$ with mutual distances $ \geqslant r$ relative to the metric $ d(A,B) = \left\Vert {A - B} \right\Vert$. We give an elementary, largely geometrical proof of the following results of Bieberbach: Let G be a subgroup of $ {E_n}$. (1) If G is discrete, then $ {G_{1/2}}$ is abelian, $ {G_{1/2}} \triangleleft G$, and $ [G:{G_{1/2}}] \leqslant {k_n}(1/2)$. (2) G is discrete if and only if $ G \subset {O_{n - k}} \times {E_k}$, where $ {p_2}G$ is discrete, $ {({p_2}G)^ \ast }$ spans $ {{\mathbf{R}}^k}$, and $ G \cap \ker {p_2}$ is finite. (Here $ {p_2}$ is the projection on the second factor.) (3) G is crystallographic if and only if G is discrete and $ {G^ \ast }$ spans $ {{\mathbf{R}}^n}$. Moreover, if G is crystallographic, then $ [G:{G^ \ast }] \leqslant {k_n}(1/2)$.


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  • [1] L. Auslander, An account of the theory of crystallographic groups, Proc. Amer. Math. Soc. 16 (1965), 1230-1236. MR 32 #2482. MR 0185012 (32:2482)
  • [2] L. Bieberbach, Über einen Satz des Hrn. C. Jordan in der Theorie der endlichen Gruppen linearer Substitutionen, S.-B. Preuss. Akad. Wiss. (1911), 231-240.
  • [3] -, Über die Bewegungsgruppen der Euklidischen Räume. I, Math. Ann. 70 (1911), 297-336. MR 1511623
  • [4] W. M. Boothby and H. C. Wang, On the finite subgroups of connected Lie groups, Comment. Math. Helv. 39 (1965), 281-294. MR 31 #4856. MR 0180622 (31:4856)
  • [5] B. N. Delone and M. I. Štogrin, Simplified proof of the Schönflies theorem, Dokl. Akad. Nauk SSSR 219 (1974), 95-98 = Soviet Physics Dokl. 19 (1975), 727-729. MR 0370329 (51:6556)
  • [6] G. Frobenius, Über die unzerlegbaren diskreten Bewegungsgruppen, S.-B. Preuss. Akad. Wiss. (1911), 654-665.
  • [7] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I, Interscience, New York and London, 1963. MR 0152974 (27:2945)
  • [8] W. Rinow, Die innere Geometrie der metrischen Räume, Die Grundlehren der mathematischen Wissenschaften, Band 105, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0123969 (23:A1290)
  • [9] K. Rohn, Krystallstruktur und regelmässige Punktgruppen, Ber. Sächs. Akad. Wiss. 51 (1899), 445-455.
  • [10] -, Einige Sätze über regelmässige Punktgruppen, Math. Ann. 53 (1900), 440-449. MR 1511096
  • [11] È. B. Vinberg, On the Schönflies-Bieberbach theorem, Dokl. Akad. Nauk SSSR 221 (1975), 1013-1015 = Soviet Math. Dokl. 16 (1975), 440-442. MR 0486187 (58:5964)
  • [12] J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. MR 36 #829. MR 0217740 (36:829)
  • [13] H. Zassenhaus, Beweis eines Satzes über diskrete Gruppen, Abh. Math. Sem. Univ. Hamburg 12 (1938), 289-312.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0574501-7
Keywords: Discrete euclidean groups, crystallographic groups
Article copyright: © Copyright 1980 American Mathematical Society

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