I would like to understand the connection between Riemann surfaces and homogeneous spaces, in the case of the upper half plane.
I am reading a book in which they define a linear element in a Riemann surface as a pair made of a point and a direction (I think of a it as a tangent vector at that point). $PGL_2(\mathbb{R})=G$ is the set of the conformal transformations (action by fractional linear transformations) of the upper half plane.
An homogeneous space is a space endowed with a continuous action, i.e. a representation of topological spaces. It can be identified with $\Gamma \backslash G$ where $\Gamma$ is a closed subgroup (namely, the stabilizer of a point).
They then interpret the homogeneous space as the space of linear elements, and say these linear elements are more interesting than the Riemann surface itself, since it is an homogeneous space. But wasn't it already the case of $\Gamma \backslash G$? Is there a subtlety I am missing here?