Geometriae Dedicata

, Volume 137, Issue 1, pp 27–61 | Cite as

Sous-groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation

  • Masseye Gaye
Original Paper


Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane \({{\mathbb{H}}^2_{\mathbb{C}}}\) and let G n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in \({{{\mathbb{H}}^2_{\mathbb{C}}}}\) . Under certain conditions, we prove that G n is discrete. We construct representations ρ of the fundamental group Γ g of the compact surface Σ g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.


Complex reflection Discrete sub-group Surface fundamental group Representation of the fundamental group of the compact surface Toledo’s invariant Representation manifold Deformation 

Mathematics Subject Classification (2000)

32Q05 20H10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ana’in, S., Grossi, C.H., Gusevskii, N.: Complex hyperbolic structures on disc bundles over surfaces. PreprintGoogle Scholar
  2. 2.
    Beardon A.: The geometry of discrete groups. Springer, New York (1983)MATHGoogle Scholar
  3. 3.
    Domic A., Toledo D.: The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276, 425–432 (1987)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Falbel E., Zocca V.: A Poincaré’s polyhedron theorem for complex hyperbolic geometry. J. Reine Angew. Math. 516, 133–158 (1999)MATHMathSciNetGoogle Scholar
  5. 5.
    Gaye, M.: Sous groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation. ThèseGoogle Scholar
  6. 6.
    Goldman, W.M.: Complex hyperbolic geometry, xx + 316 pp. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1999)Google Scholar
  7. 7.
    Goldman W.M., Kapovich M., Leeb B.: Complex hyperbolic manifolds homotopy equivalent to a Riemann surface. Commun. Anal. Geom. 9(1), 61–95 (2001)MATHMathSciNetGoogle Scholar
  8. 8.
    Gromov, M., Lawson, H.B. Jr., Thurston, W.: Hyperbolic 4-manifolds and conformally flat 3-manifolds. Inst. Hautes Études Sci. Publ. Math. No. 68 (1988), pp. 27–45 (1989)Google Scholar
  9. 9.
    Guichard O.: Groupes plongés quasi isométriquement dans un groupe de Lie. Math. Ann. 330, 331–351 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kapovich M.: Flat conformal structures on 3-manifolds, I: Uniformization of closed Seifert manifolds. J. Differ. Geom. 38(1), 191–215 (1993)MATHMathSciNetGoogle Scholar
  11. 11.
    Kuiper, N.H.: Hyperbolic 4-manifolds and tesselations. Inst. Hautes Études Sci. Publ. Math. No. 68 (1988) 47–76 (1989)Google Scholar
  12. 12.
    Luo F.: Constructing conformally flat structures on some Seifert fibred 3-manifolds. Math. Ann. 294(3), 449–456 (1992)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Maskit, B.: Kleinian Groups. Springer-Verlag (1988)Google Scholar
  14. 14.
    Toledo D.: Representations of surface groups in complex hyperbolic space. J. Differ. Geom. 29(1), 125–133 (1989)MATHMathSciNetGoogle Scholar
  15. 15.
    Xia E.Z.: The moduli of flat PU(2,1) structures on Riemann surfaces. Pcific J. Math. 195(1), 231–256 (2000)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Èquipe Analyse AlgébriqueInstitut de Mathématiques de JussieuParisFrance

Personalised recommendations