I have the following questiom: let $X$ and $Y$ be two different points (represented by Riemann surfaces) in the Teichmuller space $T_g$ of genus $g \geq 2$ Riemann surfaces. Then of course $X$ and $Y$ are homeomorphic and not bi-holomorphically equivalent. My question is, whether there exists a holomorphic covering from $X$ to $Y.$ Namely, is there a topological covering $p: X \to Y$ which is holomorphic with respec to the complex structures of $X$ and $Y$? Why or why not?
Thanks in advance!
As $g\geq 2$, it follows by Riemann-Hurwitz that any topological covering $X\rightarrow Y$ is a homeomorphism, and any holomorphic homeomorphism is biholomorphic.
As Adam said in his answer, by Riemann Hurwitz, if the genus is greater than 1, every holomorphic covering must be biholomorphic. But biholomorphic maps certainly exist. (Contrary to what is stated in the question, different points of the Teichmuller space can be bihilomorphically equivalent).
They form a group acting on the Teichmuller space called the Modular group, which acts on the Teichmuller space (and in general has fixed points). The factor over this group is called the moduli space, and the group itself has been very well studied.
Moreover, by the same reason there is no ramified covering $f\colon X\to Y$ either.
