Mobius transformations on R~3

摘要

We describe the subgroup of all quaternionic Mobius transformations q H (aq b)(cq + , 1) E SL2(H), which map R3 realized as the imaginary space of quaternions onto itself. This group acts threefold transitively and the action of quadruples of points can be described with the help of the quaternionic cross ratio. At first we describe the behaviour of the cross ratio under permutation of the points. Then we derive a necessary and sufficient condition for such a MObius transformation to possess a fixed point in R3 by an inequality in the entries of the matrix. Moreover, we classify those MObius transformations with a fixed point under conjugacy.