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
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 annulus, $n=0$ (there are only rotations)
In the case of the torus, 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.