The question is pretty much stated in the title. This is clearly true for the sphere, but I don't know anything about other cases.
2026-04-05 19:12:09.1775416329
Does the biholomorphisms group act transitively on a compact Riemann surface?
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For the compact Riemann surface of genus 0, the biholomorphism group is transitive as you already noticed. So is for genus 1 surfaces, as they have structures of complex Lie groups. They are so-called elliptic curves.
On the other hand, if genus is at least 2, it is known that the biholomorphism group is a finite group. Therefore, its action is far from transitive. A general bound from above of the order of this group is explicitly known and is called the Hurwitz bound. There are many many detailed results on these biholomorphism groups from classical to modern. Try searching online if you are interested.