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Geometriae Dedicata

, Volume 177, Issue 1, pp 43–60 | Cite as

On the isometry group of a compact flat orbifold

  • John G. Ratcliffe
  • Steven T. Tschantz
Original Paper

Abstract

Let \(\Gamma \) be an \(n\)-dimensional crystallographic group. We prove that the group \(\mathrm{Isom}(E^n/\Gamma )\) of isometries of the flat orbifold \(E^n/\Gamma \) is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of the group \(\Gamma \). This implies that \(\mathrm{Isom}(E^n/\Gamma )\) is finite if and only if \(\Gamma /[\Gamma , \Gamma ]\) is finite. We also generalize known results on the Nielsen realization problem for torsion-free \(\Gamma \) to arbitrary \(\Gamma \).

Keywords

Flat orbifold Isometry group Lie group Crystallographic group  Nielsen realization problem 

References

  1. 1.
    Adams, J.F.: Lectures on Lie Groups. W. A. Benjamin, New York (1969)MATHGoogle Scholar
  2. 2.
    Brown, H., Neubüser, J., Zassenhaus, H.: On integral groups. III: normalizers. Math. Comp. 27, 167–182 (1973)Google Scholar
  3. 3.
    Charlap, L.S., Vasquez, A.T.: Compact flat Riemannian manifolds III: the group of affinities. Am. J. Math. 95, 471–494 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Farkas, D.R.: Crystallographic groups and their mathematics. Rocky Mountain J. Math. 11, 511–551 (1981)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gubler, M.: Normalizer groups and automorphism groups of symmetry groups. Z. Krist. 158, 1–26 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Holt, D.F.: An interpretation of the cohomology groups \(H^n(G, M)\). J. Algebra 60, 307–320 (1979)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lee, K.B.: Geometric realization of \(\pi _0\varepsilon (M)\). Proc. Am. Math. Soc. 86, 353–357 (1982)MATHGoogle Scholar
  8. 8.
    Lee, K.B., Raymond, F.: Topological, affine and isometric actions on flat Riemannian manifolds I. J. Differ. Geom. 16, 255–269 (1981)MathSciNetMATHGoogle Scholar
  9. 9.
    Lee, K.B., Raymond, F.: Topological, affine and isometric actions on flat Riemannian manifolds II. Topol. Appl. 13, 295–310 (1982)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mac Lane, S.: Homology. Springer, New York (1967)Google Scholar
  11. 11.
    Mac Lane, S., Whitehead, J.H.C.: On the 3-type of a complex. Proc. Nat. Acad. Sci. USA 30, 41–48 (1956)Google Scholar
  12. 12.
    Ratcliffe, J.G.: Crossed extensions. Trans. Am. Math. Soc. 257, 73–89 (1980)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ratcliffe, J.G.: Ratcliffe Foundations of Hyperbolic Manifolds, Second Edition, Graduate Texts in Math., vol. 149. Springer, Berlin (2006)Google Scholar
  14. 14.
    Ratcliffe, J.G., Tschantz, S.T.: Abelianization of space groups. Acta Cryst. A65, 18–27 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ratcliffe, J.G., Tschantz, S.T.: Fibered orbifolds and crystallographic groups. Algebr. Geom. Topol. 10, 1627–1664 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Szczepański, A.: Holonomy groups of crystallographic groups with finite outer automorphism groups, Recent advances in group theory and low dimensional topology. In: Mennicke, J., Cho, J.R. (eds.) Proceedings of the German-Korea Workshop, Pusan 2000, Res. Exp. Math. 27. Heldermann Verlag, Lemgo, pp. 163–165 (2003)Google Scholar
  17. 17.
    Szczepański, A.: Geometry of Crystallographic Groups, Algebra and Discrete Math, vol. 4. Word Scientific, Singapore (2012)CrossRefGoogle Scholar
  18. 18.
    Zieschang, H., Zimmermann, B.: Endliche Gruppen von Abbildungsklassen gefaserter 3-Mannigfal tigkeiten. Math. Ann. 240, 41–62 (1979)Google Scholar
  19. 19.
    Zimmermann, B.: Über Gruppen von Homöomorphismen Seifertscher Faserräume und Flacher Mannigfaltigkeiten. Manuscripta Math. 30, 361–373 (1980)CrossRefMATHGoogle Scholar
  20. 20.
    Zimmermann, B.: Nielsensche Realisierungssätze für Seifertfaserungen. Manuscripta Math. 51, 225–242 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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