Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-08-30T04:59:01.021Z Has data issue: false hasContentIssue false
  • English
  • Français

JØRGENSEN’S INEQUALITY FOR QUATERNIONIC HYPERBOLIC SPACE WITH ELLIPTIC ELEMENTS

Part of: Riemann surfaces

Published online by Cambridge University Press:  02 October 2009

WENSHENG CAO  and
HAIOU TAN
WENSHENG CAO*
Affiliation:
Department of Mathematics and Physics, Wuyi University, Jiangmen 529020, PR China (email: wenscao@yahoo.com.cn)
HAIOU TAN
Affiliation:
Department of Mathematics and Physics, Wuyi University, Jiangmen 529020, PR China (email: hotan@wyu.cn)
*
✉For correspondence; 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.

In this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

Keywords

quaternionic hyperbolic spaceelliptic elementJørgensen’s inequality

MSC classification

Secondary: 30F40: Kleinian groups 57S30: Discontinuous groups of transformations

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 121 - 131
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

Supported by NSFs of China (No. 10801107, No. 10671004), NSF of Guangdong Province (No. 8452902001000043) and Educational Commission of Guangdong Province (No. LYM08097). Supported by NSF of Guangdong Province (No. 8152902001000004)

References

[1]Basmajian, A. and Miner, R., ‘Discrete subgroups of complex hyperbolic motions’, Invent. Math. 131 (1998), 85–136.CrossRefGoogle Scholar
[2]Cao, W. S., ‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc. 78 (2008), 211–224.CrossRefGoogle Scholar
[3]Cao, W. S., Parker, J. R. and Wang, X. T., ‘On the classification of quaternion Möbuis transformations’, Math. Proc. Cambridge. Philos. Soc. 137 (2004), 349–361.CrossRefGoogle Scholar
[4]Chen, S. S. and Greenberg, L., ‘Hyperbolic spaces’, Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers (Academic Press, New York, 1974), pp. 49–87.CrossRefGoogle Scholar
[5]Goldman, W. M., Complex Hyperbolic Geometry, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1999).CrossRefGoogle Scholar
[6]Jiang, Y. P., Kamiya, S. and Parker, J. R., ‘Jørgensen’s inequality for complex hyperbolic space’, Geom. Dedicata 97 (2003), 55–80.CrossRefGoogle Scholar
[7]Jiang, Y. P. and Parker, J. R., ‘Uniform discreteness and Heisenberg screw motions’, Math. Z. 243 (2003), 653–669.CrossRefGoogle Scholar
[8]Jørgensen, T., ‘On discrete groups of Möbius transformations’, Amer. J. Math. 98 (1976), 739–749.CrossRefGoogle Scholar
[9]Kamiya, S., ‘Notes on non-discrete subgroups of U(1,n;F)’, Hiroshima Math. J. 13 (1983), 501–506.CrossRefGoogle Scholar
[10]Kamiya, S., ‘Notes on elements of U(1,n;C)’, Hiroshima Math. J. 21 (1991), 23–45.CrossRefGoogle Scholar
[11]Kamiya, S. and Parker, J. R., ‘Discrete subgroups of PU(2,1) with screw parabolic elements’, Math. Proc. Cambridge Philos. Soc. 144 (2008), 443–445.CrossRefGoogle Scholar
[12]Kim, D., ‘Discreteness criterions of isometric subgroups for quaternionic hyperbolic space’, Geom. Dedicata 106 (2004), 51–78.CrossRefGoogle Scholar
[13]Kim, I. and Parker, J. R., ‘Geometry of quaternionic hyperbolic manifolds’, Math. Proc. Cambridge Philos. Soc. 135 (2003), 291–320.CrossRefGoogle Scholar
[14]Parker, J. R., ‘Shimizu’s lemma for complex hyperbolic space’, Int. J. Math. 3 (1992), 291–308.CrossRefGoogle Scholar
[15]Parker, J. R., ‘Uniform discreteness and Heisenberg translations’, Math. Z. 225 (1997), 485–505.CrossRefGoogle Scholar
[16]Xie, B. H. and Jiang, Y. P., ‘Discreteness of subgroups of PU(2,1) with regular elliptic elements’, Linear Algebra Appl. 428 (2008), 732–737.CrossRefGoogle Scholar
[17]Zhang, F., ‘Quaternions and matrices of quaternions’, Linear Algebra Appl. 251 (1997), 21–57.CrossRefGoogle Scholar