Mathematische Annalen

, Volume 332, Issue 1, pp 161–176 | Cite as

The Auslander conjecture for NIL-affine crystallographic groups

  • Dietrich BurdeEmail author
  • Karel Dekimpe
  • Sandra Deschamps


Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups Γ in Aff(N)=N⋊Aut(N) acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions 1≤n≤5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.


Natural Generalization Finite Dimension Crystallographic Group Auslander Conjecture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abels, H.: Properly discontinuous groups of affine transformations: a survey. Geom. Dedicata 87, 309–333 (2001)Google Scholar
  2. 2.
    Abels, H., Margulis, G.A., Soifer, G.A.: Properly Discontinuous Groups of Affine Transformations with Orthogonal Linear Part. C. R. Acad. Sci. Paris Sér. I Math. 324(3), 253–258 (1997)Google Scholar
  3. 3.
    Abels, H., Margulis, G.A., Soifer, G.A.: On the Zariski closure of the linear part of a properly discontinuous group of affine transformations. J. Differential Geom. 60, 315–344 (2002)Google Scholar
  4. 4.
    Auslander, L.: The structure of complete locally affine manifolds. Topology 3, 131–139 (1964)Google Scholar
  5. 5.
    Benoist, Y.: Une nilvariété non affine. J. Differential Geom. 41, 21–52 (1995)Google Scholar
  6. 6.
    Burde, D., Grunewald, F.: Modules for certain Lie algebras of maximal class. J. Pure Appl. Algebra 99, 239–254 (1995)Google Scholar
  7. 7.
    Burde, D.: Affine structures on nilmanifolds. Int. J. Math. 7, 599–616 (1996)Google Scholar
  8. 8.
    Charlap, L.S.: Bieberbach Groups and Flat Manifolds. Universitext, Springer–Verlag New York, 1986Google Scholar
  9. 9.
    Conze, J.-P., Guivarc’h, Y.: Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris, Sér. A 278, 1083–1086 (1974)Google Scholar
  10. 10.
    Dekimpe, K.: Any virtually polycyclic group admits a NIL-affine crystallographic action. Topology 42, 821–832 (2003)Google Scholar
  11. 11.
    Dekimpe, K.: The construction of affine structures on virtually nilpotent groups. Manuscripta Math. 87, 71–88 (1995)Google Scholar
  12. 12.
    Dekimpe, K., Igodt, P.: Polycyclic-by-finite groups admit a bounded-degree polynomial structure. Invent. Math. 129(1), 121–140 (1997)Google Scholar
  13. 13.
    Kamber, F., Tondeur, P.: Flat manifolds with parallel torsion. J. Differential Geom. 2 358–389 (1968)Google Scholar
  14. 14.
    Magnin, L.: Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent Lie Algebras of Dimension ≤7 over ℂ. E-book, 1995, pp. 1–906
  15. 15.
    Milnor, J.: On fundamental groups of complete affinely flat manifolds. Adv. Math. 25, 178–187 (1977)Google Scholar
  16. 16.
    Wolf, J.A.: Spaces of constant curvature. Publish or Perish, Berkeley, CA, 1977Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dietrich Burde
    • 1
    Email author
  • Karel Dekimpe
    • 2
  • Sandra Deschamps
    • 2
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Katholieke Universiteit LeuvenKortrijkBelgium

Personalised recommendations