Abstract
The weak Bieberbach theorem states that each crystallographic group on a Euclidean space uniquely determines its translation lattice as an abstract group. Garipov proved in 2003 that the same holds for crystallographic groups on Minkowski spaces and asked whether a similar claim holds in the pseudo-Euclidean spaces ℝp,q. We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces ℝp,q with min{p, q} ≤ 2. For min{p, q} ≥ 3 we construct examples of crystallographic groups with two distinct lattices exchanged by a suitable automorphism of the group. For crystallographic groups with two distinct isomorphic pseudo-Euclidean lattices we also prove that the coranks of their intersection in these lattices can take arbitrary values greater than 2 with the exception of 4.
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Original Russian Text Copyright © 2010 Churkin V. A.
The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh.344.2008.1) and the Program “Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 2.1.1/419).
To Yuriĭ Leonidovich Ershov on the occasion of his 70th birthday.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 700–714, May–June, 2010.
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Churkin, V.A. The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces. Sib Math J 51, 557–568 (2010). https://doi.org/10.1007/s11202-010-0058-8
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DOI: https://doi.org/10.1007/s11202-010-0058-8