# holomorphic covering between points in Teichmuller space

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?

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@silktomath: If you like one of the answers, you should click the "accept" button. If you like several of the answers, you should click the earliest one that you like. –  Lee Mosher Jan 19 at 15:32
Lee, there is a very good chance I first heard about Teichmuller Space from you. –  Adam Epstein Jan 19 at 16:52
Could be.. Could be. :-) –  Lee Mosher Jan 20 at 0:52

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.

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Thanks a lot for this! –  silktomath Jan 19 at 12:02
To express this a little differently, this holds for $X,Y$ if and only if they are in the same orbit of the mapping class group on $T_g$. –  Lee Mosher Jan 19 at 13:30
Moreover, by the same reason there is no ramified covering $f\colon X\to Y$ either.