Proceedings of the Japan Academy, Series A, Mathematical Sciences

A discrete criterion in $\mathit{PU}(2,1)$ by use of elliptic elements

Ainong Fang and Shihai Yang

Full-text: Open access

Abstract

In this paper we show a $2$-dimensional subgroup in $\mathit{PU}(2,1)$ which contains elliptics is discrete if and only if all its subgroups generated by two elliptics are discrete. This generalize the well-known discreteness criterion first established by T. Jørgensen.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 82, Number 3 (2006), 46-48.

Dates
First available in Project Euclid: 4 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1144158992

Digital Object Identifier
doi:10.3792/pjaa.82.46

Mathematical Reviews number (MathSciNet)
MR2214773

Zentralblatt MATH identifier
1141.20307

Subjects
Primary: 30F40: Kleinian groups [See also 20H10] 30C60 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Keywords
Discrete groups complex hyperbolic space elliptic elements

Citation

Yang, Shihai; Fang, Ainong. A discrete criterion in $\mathit{PU}(2,1)$ by use of elliptic elements. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 3, 46--48. doi:10.3792/pjaa.82.46. https://projecteuclid.org/euclid.pja/1144158992


Export citation

References

  • W. Abikoff and A. Haas, Nondiscrete groups of hyperbolic motions, Bull. London Math. Soc. 22 (1990), no. 3, 233–238.
  • A. Basmajian and R. Miner, Discrete subgroups of complex hyperbolic motions, Invent. Math. 131 (1998), no. 1, 85–136.
  • B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995), no. 1, 229–274.
  • S. S. Chen and L. Greenberg, Hyperbolic spaces, in Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 49–87.
  • M. Chen, Discreteness and convergence of Möbius groups, Geom. Dedicate 104 (2004), 61–69.
  • A. Fang and B. Nai, On the discreteness and convergence in $n$-dimensional Möbius groups, J. London Math. Soc. (2) 61 (2000), no. 3, 761–773.
  • J. Gilman, Inequalities in discrete subgroups of $\mathrm{PSL}(2,\mathbf{R})$, Canad. J. Math. 40 (1988), no. 1, 115–130.
  • W. M. Goldman, Complex hyperbolic geometry, Oxford Univ. Press, New York, 1999.
  • N. A. Isachenko, Systems of generators of subgroups of $\mathrm{PSL}(2,C)$. (Russian) Sibirsk. Mat. Zh. 31 (1990), no. 1, 191–193, 223; translation in Siberian Math. J. 31 (1990), no. 1, 162–165.
  • Y. Jiang, S. Kamiya and J. Parker, Jørgensen's inequality for complex hyperbolic space, Geom. Dedicata 97 (2003), 55–80.
  • Y. Jiang, S. Kamiya and J. R. Parker, Uniform discreteness and Heisenberg screw motions, Math. Z. 243 (2003), no. 4, 653–669.
  • T. Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749.
  • G. J. Martin, On discrete Möbius groups in all dimensions: a generalization of Jørgensen's inequality, Acta Math. 163 (1989), no. 3-4, 253–289.
  • G. J. Martin and R. K. Skora, Group actions of the $2$-sphere, Amer. J. Math. 111 (1989), no. 3, 387–402.
  • J. R. Parker, Shimizu's lemma for complex hyperbolic space, Internat. J. Math. 3 (1992), no. 2, 291–308.
  • J. R. Parker, Uniform discreteness and Heisenberg translations, Math. Z. 225 (1997), no. 3, 485–505.
  • P. Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157–187.