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

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Abstract

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.

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References

  1. Ana’in, S., Grossi, C.H., Gusevskii, N.: Complex hyperbolic structures on disc bundles over surfaces. Preprint

  2. Beardon A.: The geometry of discrete groups. Springer, New York (1983)

    MATH  Google Scholar 

  3. Domic A., Toledo D.: The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276, 425–432 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Falbel E., Zocca V.: A Poincaré’s polyhedron theorem for complex hyperbolic geometry. J. Reine Angew. Math. 516, 133–158 (1999)

    MATH  MathSciNet  Google Scholar 

  5. Gaye, M.: Sous groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation. Thèse

  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)

  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)

    MATH  MathSciNet  Google Scholar 

  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)

  9. Guichard O.: Groupes plongés quasi isométriquement dans un groupe de Lie. Math. Ann. 330, 331–351 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kapovich M.: Flat conformal structures on 3-manifolds, I: Uniformization of closed Seifert manifolds. J. Differ. Geom. 38(1), 191–215 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Kuiper, N.H.: Hyperbolic 4-manifolds and tesselations. Inst. Hautes Études Sci. Publ. Math. No. 68 (1988) 47–76 (1989)

  12. Luo F.: Constructing conformally flat structures on some Seifert fibred 3-manifolds. Math. Ann. 294(3), 449–456 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Maskit, B.: Kleinian Groups. Springer-Verlag (1988)

  14. Toledo D.: Representations of surface groups in complex hyperbolic space. J. Differ. Geom. 29(1), 125–133 (1989)

    MATH  MathSciNet  Google Scholar 

  15. Xia E.Z.: The moduli of flat PU(2,1) structures on Riemann surfaces. Pcific J. Math. 195(1), 231–256 (2000)

    Article  MATH  Google Scholar 

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  1. Èquipe Analyse Algébrique, Institut de Mathématiques de Jussieu, 75251, Paris, France

    Masseye Gaye

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  1. Masseye Gaye
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Gaye, M. Sous-groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation. Geom Dedicata 137, 27–61 (2008). https://doi.org/10.1007/s10711-008-9285-6

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  • DOI: https://doi.org/10.1007/s10711-008-9285-6

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