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Zbl 1121.30022
Cao, Wensheng
On the classification of four-dimensional Möbius transformations.
(English)
[J] Proc. Edinb. Math. Soc., II. Ser. 50, No. 1, 49-62 (2007). ISSN 0013-0915; ISSN 1464-3839/e

Ahlfors dealt in 1985 with Möbius transformations in the quaternions using matrices with quaternionic entries. The author calls such a transformation {\it parabolic} if the norms of its right eigenvalues are 1 and is has exactly one fixed point in $\hat{\mathbb{H}}=\partial {\text{\bf H}}$ with the hyperbolic 5-space {\bf H}. A transformation is called loxodromic if the norm of one of its right eigenvalues is bigger than 1. And at last, it is called elliptic if the norms of its right eigenvalues are 1 and it has at least two fixed points. Then the Möbius transformations are classified by conditions for the coefficient matrices which are too technical to be cited here. Some examples are given.
[Klaus Habetha (Aachen)]
MSC 2000:
*30G35 Functions of hypercomplex variables and generalized variables
30F40 Kleinian groups
22E40 Discrete subgroups of Lie groups
20H10 Fuchsian groups and their generalizations (group theory)

Keywords: quaternionic transformations; Möbius transformations; classification; hyperbolic 5-space

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