Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T12:35:49.651Z Has data issue: false hasContentIssue false

ALGEBRAIC CONVERGENCE THEOREMS OF COMPLEX KLEINIAN GROUPS

Published online by Cambridge University Press:  02 August 2012

WENSHENG CAO*
Affiliation:
School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, P.R. China e-mail: wenscao@yahoo.com.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Gr,i} be a sequence of r-generator subgroups of U(1,n; ℂ) and Gr be its algebraic limit group. In this paper, two algebraic convergence theorems concerning {Gr,i} and Gr are obtained. Our results are generalisations of their counterparts in the n-dimensional sense-preserving Möbius group.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Apanasov, B. N., Conformal geometry of discrete groups and manifolds (Walter de Gruyter, Berlin, Germany, 2000).Google Scholar
2.Cao, W., Discrete and dense subgroups acting on complex hyperbolic space, Bull. Aust. Math. Soc. 82 (2008), 211224.Google Scholar
3.Cao, W. and Wang, X., Discreteness criteria and algebraic convergence theorem for subgroups in PU(1, n; ℂ), Proc. Japan Acad. 82 (2006), 4952.Google Scholar
4.Chen, S. and Greenberg, L., Hyperbolic spaces, in Contributions to analysis (Academic Press, New York, 1974), 4987.Google Scholar
5.Dai, B., Fang, A. and Nai, B., Discreteness criteria for subgroups in complex hyperbolic space, Proc. Japan Acad. 77 (2001), 168172.Google Scholar
6.Goldman, W. M., Complex hyperbolic geometry (Oxford University Press, New York, 1999).Google Scholar
7.Jørgensen, T. and Klein, P., Algebraic convergence of finitely generated Kleinian groups, Quart. J. Math. 33 (1982), 325332.Google Scholar
8.Kamiya, S., Notes on elements of U(1, n; ℂ), Hiroshima Math. J. 21 (1991), 2345.Google Scholar
9.Martin, G. J., On discrete Möbius groups in all dimensions, Acta Math. 163 (1989), 253289.Google Scholar
10.Navarrete, J. P., On the limit set of discrete subgroups of PU(2, 1), Geometriae Dedicata 122 (2006), 113.Google Scholar
11.Wang, X., Algebraic convergence theorems of n-dimensional Kleinian groups, Isr. J. Math. 162 (2007), 221233.Google Scholar
12.Wang, X. and Yang, W., Discreteness criteria of Möbius groups of high dimensions and convergence theorem of Kleinian groups, Adv. Math. 159 (2001), 6882.Google Scholar