On the transitivity of the group of automorphisms of a Riemann surface Let $S$ be a Riemann surface. What can be said of the greatest integer $n$ such that the group of biholomorphisms of $S$, $\mathrm{Aut}(S)$, acts $n$-transitively on $S$ ? 
(for the Riemann sphere, it is 3 for instance)
In particular, is there any easy way to see it is always greater than one ?
Edit : by Riemann surface, I mean connected complex holomorphic 1-dimensional manifold
 A: I just remarked this : if $M$ is a hyperbolic Riemann surface, then $\mathrm{Aut}(M)$ cannot act 2-transitively on $M$. 
Indeed, if $M$ is hyperbolic then its automorphisms are isometries for the hyperbolic metric, and so map couple of points to couple of points at the same distance. So $n=0$ or $1$ for all hyperbolic Riemann surfaces. 
Besides, the non-hyperbolic Riemann surfaces are the plane, the disk, the sphere, the annuli and the torii. 


*

*In the case of the sphere, $n=3$

*In the case of the plane, $n=2$

*In the case of the disk, $n=0$

*In the case of the annuli, $n=0$ (there are only rotations)

*In the case of the torii, the automorphisms are the translations and the multiplications 
by units of the ring $\mathbb{Z}+ \tau \mathbb{Z}$. So $n \geq 1$ and I'm not sure yet if there is equality.
A: If the automorphism group of a Riemann surface is transitive, the it must be of uncountable size since for any point p there must exist a distinct group element g(p,q) to carry it to each point q.
For a compact Riemann surface M_g of genus g >= 2, the size of the automorphism group Aut(M_g) is bounded above by a theorem of Hurwitz:
|Aut(M_g)|  <=  84(g-1).
This shows Aut(M_g) cannot be transitive for g >= 2. 
Also, even though the automorphism group of the disk is 1-transitive, as mentioned above ... but if Aut(D) is extended to act on S^1 = ∂D, then the action restricted to S^1 is in fact 3-transitive.
