The linear part of an affine group acting properly discontinuously and leaving a quadratic form invariant

H. Abels, G. A. Margulis, G. A. Soifer

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12 Scopus citations


In this paper we study the dynamics of properly discontinuous and crystallographic affine groups leaving a quadratic from of signature (p, q) invariant. The main results are: (I) If p - q ≥ 2, then the linear part of the group is not Zariski dense in the corresponding orthogonal group. (II) If q = 2 and the group is crystallographic, then the group is virtually solvable. This proves the Auslander conjecture for this case.

Original languageEnglish
Pages (from-to)1-46
Number of pages46
JournalGeometriae Dedicata
Issue number1
StatePublished - Aug 2011

Bibliographical note

Funding Information:
The authors would like to thank E.B. Vinberg for many helpful discussions, several institutions and foundations for their support during the preparation of this paper: Bie-lefeld University, the Emmy Noether Institute Bar-Ilan University, Yale University, For-schergruppe “Spektrale Analysis, asymptotische Verteilungen und stochastische Dynam-ik”, SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”, NSF under grant DMS 0244406, USA-Israel Binational Science foundation under BSF grant 2004010, German–Israeli Foundation for Scientific Research and Development under GIF grant G–454–213.06/95, the Center of Excellence, under ISF grant number 8002/1, ISF grant number 657/09. Without all these supports, the paper whose authors live on three different continents could not have seen the light of day.


  • Affine group
  • Crystallographic group
  • Virtually solvable group


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