RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Sibirsk. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 1 scientific paper (total in 1 paper)

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.

Full text: PDF file (345 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 512.865.3
Received: 28.01.2010

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}


Linking options:
  • http://mi.mathnet.ru/eng/smj2119
  • http://mi.mathnet.ru/eng/smj/v51/i3/p700

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  mathscinet  isi
  • Сибирский математический журнал Siberian Mathematical Journal
    Number of views:
    This page:238
    Full text:49
    References:30
    First page:6

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019