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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: xtwang@hunnu.edu.cn
Shouyao Xiong
Affiliation:
Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan 410000, People's Republic of China
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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

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