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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Abstract

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
Volume153
Issue number1
DOIs
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.

Funding

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.

FundersFunder number
Center of Excellence
Emmy Noether Institute Bar-Ilan University
National Science FoundationDMS 0244406
Yale University
German-Israeli Foundation for Scientific Research and DevelopmentG–454–213.06/95
United States-Israel Binational Science Foundation2004010
Israel Science Foundation8002/1, 657/09

    Keywords

    • Affine group
    • Crystallographic group
    • Virtually solvable group

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