Conformal automorphism of Riemann sphere equals fractional linear transformation

This article describes the computation of the conformal automorphism group of a domain or a Riemann surface
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Statement

Any conformal automorphism of the Riemann sphere is a fractional linear transformation, i.e. a map of the form:

$z\mapsto {\frac {az+b}{cz+d}}$

Equivalently, the conformal automorphism group of the Riemann sphere is precisely the group of fractional linear transformations: namely, $PSL(2,\mathbb {C} )$.

Proof

The proof involves the following steps:

• We first observe that the conformal automorphism group acts transitively on the Riemann sphere. In particular, any point $z_{0}\in \mathbb {C}$ can be mapped to $\infty$ can be sent to $\infty$ using the map:

$z\mapsto {\frac {1}{z-z_{0}}}$

• The isotropy group at $\infty$ must comprise maps of the form $z\mapsto az+b$, because any conformal automorphism of the Riemann sphere that fixes $\infty$ must be a conformal automorphism of $\mathbb {C}$, and any conformal automorphism of the complex numbers must be an affine map.