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 language | English |
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Pages (from-to) | 1-46 |
Number of pages | 46 |
Journal | Geometriae Dedicata |
Volume | 153 |
Issue number | 1 |
DOIs | |
State | Published - 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.
Funders | Funder number |
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Center of Excellence | |
Emmy Noether Institute Bar-Ilan University | |
National Science Foundation | DMS 0244406 |
Yale University | |
German-Israeli Foundation for Scientific Research and Development | G–454–213.06/95 |
United States-Israel Binational Science Foundation | 2004010 |
Israel Science Foundation | 8002/1, 657/09 |
Keywords
- Affine group
- Crystallographic group
- Virtually solvable group