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!



Uniformization of a plane minus cantor set
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 .


