Geometriae Dedicata

, Volume 126, Issue 1, pp 155–175 | Cite as

Fundamental domains in Lorentzian geometry

Original Paper

Abstract

We consider a discrete subgroup Γ of the simply connected Lie group \(\widetilde{\operatorname{SU}}(1,1)\) of finite level, i.e. the subgroup intersects the centre of \(\widetilde{\operatorname{SU}}(1,1)\) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group \(\widetilde{\operatorname{SU}}(1,1)\). The discrete subgroup Γ acts on \(\widetilde{\operatorname{SU}}(1,1)\) by left translations. We describe the Lorentz space form \(\widetilde{\operatorname{SU}}(1,1)/\Gamma\) by constructing a fundamental domain F for Γ. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Γ satisfying the following condition: The image \(\bar\Gamma\) of Γ in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Γ. The construction depends on the group Γ and on the orbit Γ(u) of the fixed point u.

Keywords

Lorentz space form Polyhedral fundamental domain Quasihomogeneous singularity Arnold singularity series 

Mathematics Subject Classifications (2000)

53C50 14J17 32S25 51M20 52B10 

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References

  1. 1.
    Balke, L., Kaess, A., Neuschäfer, U., Rothenhäusler, F., Scheidt, S.: Polyhedral fundamental domains for discrete subgroups of PSL\((2,\mathbb R)\), Topology 37 (1998), 1247–1264.Google Scholar
  2. 2.
    Brieskorn, E., Pratoussevitch, A., Rothenhäusler, F.: The Combinatorial Geometry of Singularities and Arnold’s Series E, Z, Q, Moscow Math. J. 3(2), 273–333 (2003), the special issue dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.Google Scholar
  3. 3.
    Boissonnat, J.-D., Yvinec, M.: Algorithmic geometry. Cambridge University Press, Cambridge (1998) Translated from the 1995 French original by Hervé Brönnimann.Google Scholar
  4. 4.
    Barbot T., Zeghib, A.: Group actions on Lorentz spaces, mathematical aspects: a survey, The Einstein equations and the large scale behavior of gravitational fields, pp. 401–439. Birkhäuser, Basel (2004)Google Scholar
  5. 5.
    Coxeter H.S.M. (1969). Introduction to geometry, 2nd edn. Wiley, New York MATHGoogle Scholar
  6. 6.
    Drumm, T.A., Goldman, W.M.: Crooked planes. Electron. Res. Announc. Am. Math. Soc. 1(1), 10–17 (1995) (electronic)Google Scholar
  7. 7.
    Drumm T.A. and Goldman W.M. (1999). The geometry of crooked planes. Topology 38(2): 323–351 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dolgachev I.V. (1983). On the link space of a Gorenstein quasihomogeneous surface singularity. Math. Ann. 265: 529–540 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fischer, T.: Totalgeodätische Polytope als Fundamentalbereiche von Bewegungsgruppen der dreidimensionalen Minkowskischen Pseudosphäre, Ph.D. thesis, Universität Bonn (1992)Google Scholar
  10. 10.
    Frances C. (2005). Lorentzian kleinian groups. Comment. Math. Helv. 80: 883–910 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ghys É. (1987). Flots d’Anosov dont les feuilletages stables sont différentiables. Ann. Sci. École Norm. Sup. 20(4): 251–270 MATHMathSciNetGoogle Scholar
  12. 12.
    Goldman W.M. (1985). Nonstandard Lorentz space forms. J. Differ. Geom. 21(2): 301–308 MATHGoogle Scholar
  13. 13.
    Käss, A., Neuschäfer U., Rothenhäusler F., Scheidt S.: Fundamentalbereiche diskreter, cokompakter Untergruppen von PSU(1,1), Diploma thesis, Universität Bonn, Bonn (1996) (joint work combining four Diploma thesises)Google Scholar
  14. 14.
    Kulkarni R.S. and Raymond F. (1985). 3-dimensional Lorentz space forms and Seifert fiber spaces. J. Differ. Geom. 21: 231–268 MATHMathSciNetGoogle Scholar
  15. 15.
    Milnor J.: On the 3-dimensional Brieskorn manifolds M(p,q,r). In: Neuwirth L.P. (ed.) Knots, groups and 3-manifolds, Ann. Math. Studies, vol. 84, pp. 175–225. Princeton University Press, Princeton (1975)Google Scholar
  16. 16.
    Möhring, K.: Numerical invariants and series of quasihomogeneous singularities, preprint.Google Scholar
  17. 17.
    Neumann W.D. (1977). Brieskorn complete intersections and automorphic forms. Invent. Math. 42: 285–293 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Neumann, W.D.: Geometry of quasihomogeneous surface singularities. In: Orlik P. (ed.) Proceedings of the summer Institute on Singularities Held at Humboldt State University, Arcata, Calif., 1981, Proceedings of Symposia in Pure Mathematics, vol. 40, 2, pp. 245–258. American Math. Soc., Providence (1983)Google Scholar
  19. 19.
    Pratoussevitch, A.: Polyedrische Fundamentalbereiche diskreter Untergruppen von \(\widetilde{\rm SU}(1,1)\), Bonner Mathematische Schriften [Bonn Mathematical Publications], 346, Universität Bonn, Mathematisches Institut, Bonn, 2001, Dissertation, Rheinische Friedrich- Wilhelms-Universität Bonn, Bonn (2001)Google Scholar
  20. 20.
    Pratoussevitch, A.: On the Link Space of a \(\mathbb{Q}\)-Gorenstein Quasi-Homogeneous Surface Singularity, In: Proceedings of the VIII Sao Carlos Workshop in Real and Complex Singularities in Luminy, pp. 311–325. Birkhäuser (2006)Google Scholar
  21. 21.
    Rothenhäusler, F.: Fundamentalpolyeder zu Fuchsschen Gruppen der Signatur (0,4;p,2,2,2), Bonner Mathematische Schriften [Bonn Mathematical Publications], 343, Universität Bonn, Mathematisches Institut, Bonn, 2001, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2001)Google Scholar
  22. 22.
    Salein F. (2000). Variétés anti-de Sitter de dimension 3 exotiques. Ann. Inst. Fourier (Grenoble) 50: 257–284 MATHMathSciNetGoogle Scholar

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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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