Cross-ratio-preserving bijections are Möbius

Statement

Let $F:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a bijection such that

[F(z_1),F(z_2);F(z_3),F(z_4)] =[z_1,z_2;z_3,z_4]

for every ordered quadruple of distinct points. Then $F$ is a Möbius transformation.

Proof

By the sharp three-transitivity of the Möbius group, choose a Möbius transformation $T$ agreeing with $F$ on three distinct points. For every fourth point $z$ , preservation of its cross-ratio with the chosen triple forces $F(z)=T(z)$ . Hence $F=T$ on the whole sphere.

Scope

Preservation of generalized circles alone is weaker. Without an orientation or cross-ratio condition, it also permits anti-Möbius transformations. The theorem specifically assumes preservation of the complex-valued ordered cross-ratio.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapter 3, §§1–2.