Proceedings of the Japan Academy, Series A, Mathematical Sciences

Discreteness of subgroups of $PU(1,n;\mathbf{C})$

YuePing Jiang, Hua Wang, and BaoHua Xie

Full-text: Open access

Abstract

In this paper, we discuss three discreteness criterions of $n$-dimensional subgroup $G$ of $PU(1,n;\mathbf{C})$. This generalize some discreteness criterions established by J. Gilman [3], S. Yang and A. Fang [9].

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 6 (2008), 78-80.

Dates
First available in Project Euclid: 3 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1212500760

Digital Object Identifier
doi:10.3792/pjaa.84.78

Mathematical Reviews number (MathSciNet)
MR2422734

Zentralblatt MATH identifier
1167.30024

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

Keywords
Discrete groups dense group regular elliptic elements

Citation

Xie, BaoHua; Jiang, YuePing; Wang, Hua. Discreteness of subgroups of $PU(1,n;\mathbf{C})$. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 6, 78--80. doi:10.3792/pjaa.84.78. https://projecteuclid.org/euclid.pja/1212500760


Export citation

References

  • W. Abikoff and A. Hass, Nondiscrete groups of hyperbolic motions, Bull. London Math. Soc. 22 (1990), no. 3, 233–238.
  • S. S. Chen and L. Greenberg, Hyperbolic spaces, in Contributions to analysis (a collection of papers dedicated to Lipman Bers), 49–87, Academic Press, New York, 1974.
  • J. Gilman, Inequalities in discrete subgroup of $PSL(2,R)$, Canad. J. Math. 40 (1988), no. 1, 115–130.
  • W. M. Goldman, Complex hyperbolic geometry, Oxford University Press, Oxford, New York, 1999.
  • N. A. Isachenko, Sibirsk. Mat. Zh. 31 (1990), no. 1, 191–193, 223; translation in Siberian Math. J. 31 (1990), no. 1, 162–165.
  • T. Jørgensen, A note on subgroup of $SL(2,C)$, Quart. J. Math. Oxford. 33 (1982), 325–332.
  • S. Kamiya, Notes on elements of $U(1,n;C)$, Hiroshima Math. J. 21 (1991), no. 1, 23–45.
  • S. Kamiya, On discrete subgroups of $PU(1,2;C)$ with Heisenberg translations, J. London Math. Soc. 62 (2000), no. 3, 824–842.
  • S. Yang and A. Fang, A discrete criterion in $PU(2,1)$ by use of elliptic elements, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 3, 46–48.