Möbius transformation group

Definition

The Möbius transformation group is the group of all Möbius transformations under composition. The projective action gives canonical isomorphisms

\operatorname{Möb}(\widehat{\mathbb C}) \cong PGL_2(\mathbb C) \cong PSL_2(\mathbb C),

where the second isomorphism uses the existence of square roots in $\mathbb C$ .

Action on triples

The action on the Riemann sphere is sharply three-transitive: for any two ordered triples of distinct points, there is exactly one Möbius transformation carrying the first triple to the second. The cross-ratio is the corresponding invariant of ordered quadruples.

Complex Lie group

The group has complex dimension $3$ and underlying real dimension $6$ . In the $PSL_2(\mathbb C)$ presentation it is $SL_2(\mathbb C)/\{\pm I\}$ , connecting Möbius geometry to the Lorentz and hyperbolic actions of the projective special linear group.

Conformal group of the round sphere

Under stereographic projection, the Möbius group is exactly the group of orientation-preserving conformal diffeomorphisms of the round $2$ -sphere. The full conformal diffeomorphism group also contains orientation-reversing anti-Möbius transformations, such as complex conjugation composed with a Möbius transformation.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapters 3–4.