Proceedings of the Japan Academy, Series A, Mathematical Sciences

Discreteness criteria for subgroups in complex hyperbolic space

Binlin Dai, Ainong Fang, and Bing Nai

Full-text not supplied by publisher

Abstract

In this paper, we study the discreteness criteria for subgroups of $U(1, n; \mathbf{C})$ in complex hyperbolic space $H^n_{\mathbf{C}}$. We prove that a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ with condition A is discrete if and only if every two generator subgroup of $G$ is discrete. We also prove that if a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ contains a sequence of distinct elements $\{g_m\}$ with $\operatorname{Card}(\operatorname{fix}(g_m) \cap \partial H^n_{\mathbf{C}}) \ne \infty$ and $g_m \rightarrow I$ as $m \rightarrow \infty$, then $G$ contains a non-discrete, nonelementary two generator subgroup.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 10 (2001), 168-172.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392983

Digital Object Identifier
doi:10.3792/pjaa.77.168

Mathematical Reviews number (MathSciNet)
MR1873738

Zentralblatt MATH identifier
1009.32016

Subjects
Primary: 30F40: Kleinian groups [See also 20H10] 30C62: Quasiconformal mappings in the plane 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Keywords
Complex hyperbolic space limit set elementary groups

Citation

Dai, Binlin; Fang, Ainong; Nai, Bing. Discreteness criteria for subgroups in complex hyperbolic space. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 10, 168--172. doi:10.3792/pjaa.77.168. https://projecteuclid.org/euclid.pja/1148392983


Export citation

References

  • Kamiya, S.: Notes on some classical series associated with discrete subgroups of $U(1,n;\mathbf{C})$ on $\partial{B^n}\times\partial{B^n}\times\partial{B^n}$. Proc. Japan Acad., 68A, 137–139 (1992).
  • Kamiya, S.: Chordal and matrix norms of unitary transformations. First Korean-Japanese colloquium on finite or infinite dimensional complex analysis (eds. Kajiwara, J., Kazama, H., and Shon, K. H.). pp. 121–125 (1993).
  • Kamiya, S.: On discrete subgroups of $PU(1,2;\mathbf{C})$ with Heisenberg translations. J. London Math. Soc., 62(2), 827–842 (2000).
  • Chen, S. S., and Greenberg, L.: Hyperbolic Space. Contributions to Analysis (eds. Kra, I., and Maskit, B.). Academic Press, New York, pp. 49–87 (1974).
  • Goldman, W. M.: Complex Hyperbolic Geometry. Oxford University Press, Oxford (1999).
  • Goldman, W. M., and Paker, J. R.: Dirichlet polyhedra for dihedral groups in complex hyperbolic space. J. Geom. Anal., 2, 517–554 (1992).
  • Parker, J. R.: On ford isometric spheres in complex hyperbolic space. Math. Proc. Cambridge Phil. Soc., 115, 501–512 (1994).
  • Parker, J. R.: Uniform discreteness and Heisenberg translations. Math. Z., 225, 485–505 (1997).
  • Jørgensen, T.: On discrete groups of Möbius transformations. Amer. J. Math., 98(3), 739–749 (1976).
  • Isachenko, N. A.: Systems of generators of subgroups of $PSL(2,\mathbf{C})$. Sib. Math. J., 31, 162–165 (1990).
  • Martin, G. J.: On the discrete Möbius groups in all dimensions. Acta Math., 163, 253–289 (1989).
  • Martin, G. J.: On discrete isometry groups of negative curvature. Pacific J. Math., 160(1), 109–127 (1993).
  • Abikoff, W., and Hass, A.: Non-discrete groups of hyperbolis motions. Bull. London Math. Soc., 22, 233–238 (1990).
  • Waterman, P. L.: Möbius transformations in several dimensions. Adv. Math., 101, 87–113 (1993).
  • Fang, A., and Nai, B.: On discreteness and convergence of Möbius transformation groups. J. London Math. Soc., 61(2), 761–773 (2000).
  • Jørgensen, T.: A note on subgroups of $SL(2,\mathbf{C})$. Quart. J. Math. Oxford, 28(2), 209–212 (1977).
  • Kamiya, S.: Notes on elements of $U(1,n;\mathbf{C})$. Hiroshima Math. J., 21, 23–45 (1991).
  • Martin, G. J., and Skora, R.: Group actions of the two sphere. Amer. J. Math., 111, 387–402 (1989).
  • Gehring, F. W., and Martin, G. J.: Discrete guasiconformal groups. Proc. London Math. Soc., 55(3), 331–358 (1987).
  • Bowditch, B. H.: Geometrical finiteness with variable negative curvature. Duke Math. J., 77(1), 229–274 (1995).
  • Friedland, S., and Hersonsky, S.: Jorgensen's inequality for discrete groups in normed algebras. Duke Math. J., 69, 593–614 (1993).
  • Ballmann, W., Gromov, M., and Schroeder, V.: Manifolds of Nonpositive Curvature. Progress in Mathematics, vol. 61, Birkhäuser, Boston (1985).
  • Beardon, A. F.: The Geometry of Discrete Groups. Springer, New York-Heidelberg-Berlin (1983).
  • Fang, A., Jiang, Y., and Fang, M.: On groups of Clifford matrices and Lie groups. Complex Variables, 31, 65–73 (1996).
  • Maskit, B.: Kleinian Groups. Springer, Berlin (1987).