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
View other such computations

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, $PSL(2,\mathbb {C} )$.

Related facts

• Conformal automorphism of disk implies fractional linear transformation: The technique used in this proof is the same: first, we show that the fractional linear transformations act transitively; next, we show that they contain the isotropy at $0$>
• Conformal automorphism of complex numbers implies affine map

Related techniques

• Groupprops:Proving product of subgroups: A survey article on the group theory wiki about proving that a given group is a product of two subgroups.

Facts used

1. conformal automorphism of complex numbers implies affine map

Proof

Suppose $G$ denotes the conformal automorphism group of the Riemann sphere, and $H$ the subgroup comprising fractional linear transformations. We want to show that $H=G$. We show this in two steps:

• $H$ acts transitively on the Riemann sphere.
• $H$ contains the isotropy subgroup at $\infty$.

With both these facts, observe that for any $g\in G$, we can find $h\in H$ such that $h^{-1}g$ is in the isotropy subgroup of $\infty$, which in turn is in $H$. This forces $G=H$.

Transitive action of fractional linear transformations

We first observe that the group of fractional linear transformations 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:

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

Isotropy at infinity is contained in fractional linear transformations

The isotropy group at $\infty$ must comprise maps of the form $z\mapsto az+b$, combining fact (1) with the observation that any conformal automorphism of the Riemann sphere that fixes $\infty$ must be a conformal automorphism of $\mathbb {C}$.