Acta Mathematica

, Volume 196, Issue 1, pp 1–64 | Cite as

Cone metrics on the sphere and Livné’s lattices

  • John R. ParkerEmail author


We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livné. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between these groups and other groups of complex hyperbolic isometries.


Cone Angle Complex Line Hermitian Form Cone Point Cone Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bowditch, B.H.: Geometrical finiteness with variable negative curvature. Duke Math. J., 77, 229–274 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brehm, U.: The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom. Dedicata 33, 59–76 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Deligne, P., Mostow, G.D.: Commensurabilities among Lattices in PU (1, n). Annals of Mathematics Studies, 132. Princeton University Press, Princeton, NJ (1993)Google Scholar
  4. 4.
    Deraux, M.: Deforming the R-Fuchsian (4,4,4)-triangle group into a lattice. To appear in Topology.Google Scholar
  5. 5.
    Deraux, M., Falbel, E., Paupert, J.: New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math. 194, 155–201 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Epstein, D.B.A., Petronio, C.: An exposition of Poincaré’s polyhedron theorem. Enseign. Math. 40, 113–170 (1994)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Falbel, E., Parker, J.R.: The geometry of the Eisenstein–Picard modular group. Duke Math. J. 131, 249–289 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Goldman, W.M.: Complex Hyperbolic Geometry. Oxford University Press, New York, (1999)zbMATHGoogle Scholar
  9. 9.
    Hirzebruch, F.: Arrangements of lines and algebraic surfaces. In: Arithmetic and Geometry, Vol. II., pp. 113–140. Progr. Math., 36. Birkhäuser, Boston, MA (1983)Google Scholar
  10. 10.
    Jiang, Y., Kamiya, S., Parker, J.R.: Jørgensen’s inequality for complex hyperbolic space. Geom. Dedicata 97, 55–80 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kapovich, M.: On normal subgroups in the fundamental groups of complex surfaces. Preprint, 1998Google Scholar
  12. 12.
    Livné, R.A.: On Certain Covers of the Universal Elliptic Curve. Ph.D. Thesis, Harvard University, 1981Google Scholar
  13. 13.
    Mostow, G.D.: On a remarkable class of polyhedra in complex hyperbolic space. Pacific J. Math. 86, 171–276 (1980)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Mostow, G.D.: Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math. 63, 91–106 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mostow, G.D.: On discontinuous action of monodromy groups on the complex n-ball. J. Amer. Math. Soc. 1, 555–586 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Parker, J.R.: Unfaithful complex hyperbolic triangle groups. Preprint, 2005Google Scholar
  17. 17.
    Pratoussevitch, A.: Traces in complex hyperbolic triangle groups. Geom. Dedicata 111, 159–185 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Sauter, J. K., Jr.: Isomorphisms among monodromy groups and applications to lattices in PU (1,2). Pacific J. Math. 146, 331–384 (1990)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Schwartz, R. E.: Complex hyperbolic triangle groups. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 339–349. Higher Ed. Press, Beijing, 2002Google Scholar
  20. 20.
    Schwartz, R.E.: Real hyperbolic on the outside, complex hyperbolic on the inside. Invent. Math. 151, 221–295 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Thurston, W. P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, pp. 511–549. Geom. Topol. Monogr., 1. Geom. Topol. Publ., Coventry, 1998Google Scholar
  22. 22.
    Weber, M.: Fundamentalbereiche komplex hyperbolischer Flächen. Bonner Mathematische Schriften, 254. Universit¨ at Bonn, Mathematisches Institut, Bonn, 1993Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DurhamEnglandUK

Personalised recommendations