This article is cited in 1 paper

The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces

V. A. Churkin<sup>ab

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University, Mechanics and Mathematics Department, Novosibirsk

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 $\mathbb R^{p,q}$. We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces $\mathbb R^{p,q}$ with $\min\{p,q\}\le2$. For $\min\{p,q\}\ge3$ 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.

Keywords: pseudo-Euclidean space, crystallographic group, weak Bieberbach theorem, translation lattice.

UDC: 512.865.3

Received: 28.01.2010

 English version: Siberian Mathematical Journal, 2010, 51:3, 557–568

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