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DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE

Part of: Riemann surfaces

Published online by Cambridge University Press:  01 October 2008

WENSHENG CAO
WENSHENG CAO*
Affiliation:
Department of Mathematics and Physics, Wuyi University, Jiangmen 529020, China (email: wenscao@yahoo.com.cn)
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Abstract

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In this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

Keywords

discreteness criterionnonelementary subgroupdense subgroupnormalizer

MSC classification

Secondary: 30F40: Kleinian groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 211 - 224
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Apanasov, B. N., ‘Geometry and topology of complex hyperbolic and Cauchy–Riemannian manifolds’, Russian Math. Surveys 52(5) (1997), 895–928.CrossRefGoogle Scholar
[2]Basmajian, A. and Miner, R., ‘Discrete subgroups of complex hyperbolic motions’, Invent. Math. 131(1) (1998), 85–136.CrossRefGoogle Scholar
[3]Cao, W., Parker, J. R. and Wang, X., ‘On the classification of quaternion Möbius transformations’, Math. Proc. Cambridge Philos. Soc. 137(2) (2004), 349–361.CrossRefGoogle Scholar
[4]Cao, W. and Wang, X., ‘Geometric characterizations for subgroups of PU(1,n;ℂ)’, Northeast. Math. J. 21(1) (2005), 45–53.Google Scholar
[5]Cao, W. and Wang, X., ‘Discreteness criteria and algebraic convergence theorem for subgroups in PU(1,n;ℂ)’, Proc. Japan Acad. 82(3) (2006), 49–52.Google Scholar
[6]Chen, M., ‘Discreteness and convergence of Möbius groups’, Geom. Dedicata 104 (2004), 61–69.CrossRefGoogle Scholar
[7]Chen, S. S. and Greenberg, L., ‘Hyperbolic spaces’, in: Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers (eds. L. V. Ahlfors et al.) (Academic Press, New York, 1974), pp. 49–87.Google Scholar
[8]Dai, B., Fang, A. and Nai, B., ‘Discreteness criteria for subgroups in complex hyperbolic space’, Proc. Japan Acad. 77(10) (2001), 168–172.Google Scholar
[9]Jørgensen, T., ‘On discrete groups of Möbius transformations’, Amer. J. Math. 98(3) (1976), 739–749.CrossRefGoogle Scholar
[10]Kamiya, S., ‘Notes on elements of U(1,n;ℂ)’, Hiroshima Math. J. 21(1) (1991), 23–45.Google Scholar
[11]Katok, S., Fuchsian Groups (University of Chicago Press, Chicago, 1992).Google Scholar
[12]Korányi, A. and Reimann, H. M., ‘Quasiconformal mappings on the Heisenberg group’, Invent. Math. 80(2) (1985), 309–338.Google Scholar
[13]Maskit, B., Kleinian Groups (Springer, Berlin, 1994).Google Scholar
[14]Martin, G. J., ‘On discrete Möbius groups in all dimensions’, Acta Math. 163(1) (1989), 253–289.Google Scholar
[15]Parker, J. R., ‘Uniform discreteness and Heisenberg translations’, Math. Z. 225(3) (1997), 485–505.CrossRefGoogle Scholar
[16]Ratcliffe, J., ‘On the isometry groups of hyperbolic manifolds’, Contemp. Math. 169 (1994), 491–495.Google Scholar
[17]Sullivan, D., ‘Quasiconformal homeomorphisms and dynamics: Structural stability implies hyperbolicity for Kleinian groups’, Acta Math. 155(3) (1985), 243–260.Google Scholar
[18]Wang, X., ‘Dense subgroups of n-dimensional Möbius groups’, Math. Z. 243(4) (2003), 643–651.Google Scholar
[19]Wang, X., Li, L. and Cao, W., ‘Discreteness criteria for Möbius groups acting on ’, Israel J. Math. 150 (2005), 357–368.Google Scholar
[20]Wang, X., Li, Y. and Xia, M., ‘The discreteness of the normalizers of higer dimensional Kleinian groups and the isomorphisms between Kleinian groups induced by quasiconformal mappings’, Glasg. Math. J. 47(2) (2005), 373–378.CrossRefGoogle Scholar