If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without fixed points and properly discontinuously on $\mathbb P^1$?
2026-04-03 20:56:35.1775249795
The action of automorphisms on the Riemann sphere
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If a function from the $2$-sphere to itself has no fixed points, then it is homotopic to the antipodal map. In particular it has degree $-1$.