Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T11:26:25.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Two necessary and sufficient conditions for the extension of Möbius groups

Published online by Cambridge University Press:  17 April 2009

Xiantao Wang  and
Shouyao Xiong
Xiantao Wang
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China, e-mail: [email protected]
Shouyao Xiong
Affiliation:
Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan 410000, People's Republic of China
Rights & Permissions [Opens in a new window]

Extract

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 SL(2, Γn) be the n-dimensional Clifford matrix group and G ⊂ SL(2, Γn) be a non-elementary subgroup. We show that G is the extension of a subgroup of SL(2, ℂ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exist loxodromic elements g0, h ∈ G′ such that fix(g0) = {0, ∞}, fix(g0) ∩ fix(h) = ∅ and fix(h) ∩ ℂ ≠ ∅;

(2) tr(g) ∈ ℂ for each loxodromic element g ∈ G′.

Further G is the extension of a subgroup of SL(2, ℝ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:

(1) there exists a loxodromic element g0 ∈ G′ such that fix(g0) ∩ {0, ∞} ≠ ∅;

(2) tr(g) ∈ ℝ for each loxodromic element g ∈ G′.

The discreteness of subgroups of SL(2, Γn) is also discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 29 - 35
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Ahlfors, L.V., ‘On the fixed points of Möbius transformations in ’, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15–27.CrossRefGoogle Scholar
[2]Apanasov, B.N., The geometry of discrete groups and manifolds (Nauka, Moscow, 1991).Google Scholar
[3]Beardon, A.F., The geometry of discrete groups, Graduate text in Mathematics 91 (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[4]Chen, M., ‘The extension of Möbius groups’, Complex Var. Theory Appl. 47 (2002), 225–228.Google Scholar
[5]Jørgensen, T., ‘A note on subgroups of SL(2, ℂ)’, Quart. J. Math. Oxford Ser. 2 28 (1977), 209–212.CrossRefGoogle Scholar
[6]Maski, B., Kleinian groups, Grundlehren der Mathematischen Wissenschaften 287 (Springer-Verlag, Berlin, Heidelberg, New York, 1988).Google Scholar
[7]Wang, X. and Yang, W.„ ‘Generating systems of subgroups in PSL(2, Γn)’, Proc. Edinburgh Math. Soc. 45 (2002), 49–58.CrossRefGoogle Scholar
[8]Wang, X. and Yang, W., ‘Discreteness Criteria of Mobius groups of high dimensions and convergence theorems of Kleinian groups’, Adv. Math. 159 (2001), 68–82.CrossRefGoogle Scholar
[9]Wang, X. and Yang, W., ‘Discrete criteria for subgroups in SL(2, C)’, Math. Proc. Cambridge. Philos. Soc. 124 (1998), 51–55.CrossRefGoogle Scholar