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Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set.

Question: Is there a conformal bijection between $\mathbb{D}/G$ and $\mathbb{C} \setminus K$ for some Cantor set $K \subset \mathbb{C}$?

I'm pretty sure the answer is yes but the version of the uniformization theorem I know doesn't imply this (just that the plane minus any Cantor set can be uniformized by some Fuchsian group $G$). Any good references?

Thanks in advance!

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  • $\begingroup$ Koebe Uniformisation theorem says any planar Riemann surface is biholomorphic to a domain in the Riemann sphere . $\endgroup$ Nov 10, 2014 at 17:09
  • $\begingroup$ Any good reference? $\endgroup$ Nov 10, 2014 at 17:15
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    $\begingroup$ See for example George Springer's book on Riemann surfaces $\endgroup$ Nov 10, 2014 at 17:22
  • $\begingroup$ Thanks! I'd accept this as an answer. If it were an answer... which it is. Anyway, thanks for the reference. $\endgroup$ Nov 10, 2014 at 17:24

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Koebe uniformisation theorem says that any planar Riemann surface is biholomorphic to a domain in the Riemann sphere .See George Springer's book on Riemann surfaces.At the risk of self promotion you can also look at my book with T Napier titled An Introduction to Riemann Surfaces .

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