Geometriae Dedicata

, Volume 156, Issue 1, pp 49–70 | Cite as

Configurations of flags and representations of surface groups in complex hyperbolic geometry

  • Julien Marché
  • Pierre Will
Original Paper


In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane \({\bf H^2_{\mathbb {C}}}\) . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π 1(Σ).


Complex hyperbolic geometry Representations of surface group Decorated triangulation 

Mathematics Subject Classification (2000)

22E40 32M15 51M10 57M50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Falbel E., Koseleff P.V.: Rigidity and flexibility of triangle groups in complex hyperbolic geometry. Topology 41(4), 767–786 (2002)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Falbel E., Parker J.: The moduli space of the modular group in complex hyperbolic geometry. Inv. Math. 152(1), 57–88 (2003)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Falbel E., Platis I.: The PU(2,1)-configuration space of four points in S 3 and the Cross-Ratio Variety. Math. Ann. 340(4), 935–962 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Fock V., Goncharov A.B.: Moduli spaces of convex projective structures on surfaces. Adv. Math. 208(1), 249–273 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Fock V., Goncharov A.B.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Etudes Sci. 103, 1–211 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Goldman, W.: Representations of fundamental groups of surfaces. In: Geometry and Topology (College Park, Md., 1983/84), pp. 95–117. Springer (1985)Google Scholar
  7. 7.
    Goldman W.: Complex Hyperbolic Geometry. Oxford University Press, Oxford (1999)MATHGoogle Scholar
  8. 8.
    Goldman W., Kapovich M., Leeb B.: Complex hyperbolic manifolds homotopy equivalent to a Riemann surface. Comm. Anal. Math. 9, 61–95 (2001)MATHMathSciNetGoogle Scholar
  9. 9.
    Goldman W., Parker J.: Complex hyperbolic ideal triangle groups. J. für dir reine und angewandte Math. 425, 71–86 (1992)MATHMathSciNetGoogle Scholar
  10. 10.
    Koranyi A., Reimann H.M.: The complex cross-ratio on the Heisenberg group. L’Enseign. Math. 33, 291–300 (1987)MATHMathSciNetGoogle Scholar
  11. 11.
    Parker J., Platis I.: Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47(2), 101–135 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Parker J., Platis I.: Open sets of maximal dimension in complex hyperbolic quasi-fuchsian space. J. Diff. Geom. 73, 319–350 (2006)MATHMathSciNetGoogle Scholar
  13. 13.
    Pratoussevitch A.: Traces in complex hyperbolic triangle groups. Geometriae Dedicata 111, 159–185 (2005)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Schwartz R. E.: Complex hyperbolic triangle groups. Proc. Int. Math. Cong. 1, 339–350 (2002)Google Scholar
  15. 15.
    Toledo D.: Representations of surface groups in complex hyperbolic space. J. Differ. Geom. 29, 125–133 (1989)MATHMathSciNetGoogle Scholar
  16. 16.
    Will, P.: Groupes libres, groupes triangulaires et tore épointé dans PU(2,1). Thèse de l’université Paris VI.Google Scholar
  17. 17.
    Will P.: Traces, Cross-ratios and 2-generator Subgroups of PU(2,1). Can. J. Math. 61, 1407–1436 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Will P.: The punctured torus and Lagrangian triangle groups in PU(2,1). J. reine angew. Math. 602, 95–121 (2007)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Will, P.: Bending Fuchsian representations of fundamental groups of cupsed surfaces in PU(2,1). To appear in J. differ. Geom.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Centre de Mathématiques, Laurent Schwarz, Ecole PolytechniquePalaiseauFrance
  2. 2.Institut Fourier, Université Joseph FourierSt-Martin d’HèresFrance

Personalised recommendations