On the n-transitivity of the group of Möbius transformations on C∞

Möbius transformations generate the conformal group in the plane and have been used in neural networks and conformal field theory. Some invariant characteristic properties of Möbius transformations such as the invariance of cross-ratio of four distinct points on the extended complex plane C∞=C∪{∞} under a Möbius transformation, have many applications. We consider the geometric interpretation of the notion of n-transitivity of the group of Möbius transformations on the extended complex plane C∞. We see that this notion is closely related to the invariant characteristic properties of Möbius transformations and the notion of cross-ratio.