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 Sibirsk. Mat. Zh., 2010, Volume 51, Number 3, Pages 700–714 (Mi smj2119)

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

V. A. Churkinab

a 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.

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

Bibliographic databases:

UDC: 512.865.3

Citation: V. A. Churkin, “The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces”, Sibirsk. Mat. Zh., 51:3 (2010), 700–714; Siberian Math. J., 51:3 (2010), 557–568

Citation in format AMSBIB
\Bibitem{Chu10} \by V.~A.~Churkin \paper The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces \jour Sibirsk. Mat. Zh. \yr 2010 \vol 51 \issue 3 \pages 700--714 \mathnet{http://mi.mathnet.ru/smj2119} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2722682} \zmath{https://zbmath.org/?q=an:1202.20050} \elib{http://elibrary.ru/item.asp?id=15505479} \transl \jour Siberian Math. J. \yr 2010 \vol 51 \issue 3 \pages 557--568 \crossref{https://doi.org/10.1007/s11202-010-0058-8} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279087500020} \elib{http://elibrary.ru/item.asp?id=15332580} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954026129} 

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This publication is cited in the following articles:
1. V. A. Churkin, “Crystallographic groups with two lattices and metric Lie algebras”, Algebra and Logic, 52:6 (2014), 513–516
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