I am trying to get some intuition about why the torus and the sphere are the only surfaces which can be realised as homogeneous spaces. On the one hand, I know this is true because there is the result that homogeneous spaces must have non-negative Euler characteristic:
A Structure Theorem for Homogeneous Spaces, Mostow, G, Geometriae Dedicata, (114) 2005, 87-102
However, on the other hand, a higher genus surface can be realised as a quotient of hyperbolic space by a group of isometries. The latter would seem (in my head) to give rise to a hyperbolic surface where the points look the same; all of the points have the same curvature, for instance.
My question is then: How do you distinguish the points of such a hyperbolic surface?
Best Answer
There is a classical theorem of Hurwitz which states that the group of biholomorphisms of a Riemann surface of genus $g>1$ contains at most $84(g-1)$ elements. Such a Riemann surface cannot be a homogeneous space because the (orientation preserving) isometries are conformal maps and thus they are also biholomorphic maps.
The proof is of Hurwitz' theorem is based on the concept of Weierstrass point. There are at least $2g+2$ such points on any Riemann surface of genus $g>1$, but their number is bounded from above by a universal function of $g$. The set of such points is a biholomorphism invariant. The clincher is that a biholomorphic map of a Riemann surface cannot have too many fixed points. For more details see Eric Reyssat's Quelques Aspects des Surfaces de Riemann, Birkhauser or Rick Miranda's Algebraic Curves and Riemann Surfaces, Amer. Math. Soc.