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On the Discreteness and Convergence in n ‐Dimensional Möbius Groups
Author(s) -
Aig Fang,
Bing Nai
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008784
Subject(s) - mathematics , clifford algebra , bounded function , combinatorics , torsion (gastropod) , group (periodic table) , pure mathematics , algebra over a field , physics , mathematical analysis , quantum mechanics , medicine , surgery
Throughout this paper, we adopt the same notations as in [ 1 , 6 , 8 ] such as the Möbius group M (R n ), the Clifford algebra C n −1 , the Clifford matrix group SL(2, Γ n ), the Clifford norm of A = (a bc d) ∈ S L ( 2 , Γ n ) :‖ A ‖=(| a | 2 +| b | 2 +| c | 2 +| d | 2 ) ½ (1) and the Clifford metric of SL(2, Γ n ) or of the Möbius group M (R n ) d ( A 1 , A 2 )=‖ A 1 − A 2 ‖(| a 1 − a 2 | 2 +| b 1 − b 2 | 2 +| c 1 − c 2 | 2 +| d 1 − d 2 | 2 ) ½ (2) where ∣·∣ is the norm of a Clifford number andA i = (a ib ic id i) ∈ S L ( 2 , Γ n )represents f i ∈ M (R ¯ n ), i = 1, 2, and so on. In addition, we adopt some notions in [ 6 , 12 ]: the elementary group, the uniformly bounded torsion, and so on. For example, the definition of the uniformly bounded torsion is as follows.