Acta Mathematica

Cone metrics on the sphere and Livné’s lattices

John R. Parker

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Abstract

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.

Article information

Source
Acta Math., Volume 196, Number 1 (2006), 1-64.

Dates
Received: 20 May 2005
Revised: 21 October 2005
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891804

Digital Object Identifier
doi:10.1007/s11511-006-0001-9

Mathematical Reviews number (MathSciNet)
MR2237205

Zentralblatt MATH identifier
1100.57017

Rights
2006 © Springer-Verlag

Citation

Parker, John R. Cone metrics on the sphere and Livné’s lattices. Acta Math. 196 (2006), no. 1, 1--64. doi:10.1007/s11511-006-0001-9. https://projecteuclid.org/euclid.acta/1485891804


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References

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