In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow:
A polynomail-like map of degree d is a triple $(U,U',f)$ where $U$ and $U'$ are open subsets of $\mathbb{C}$ isomorphic to discs, with $U'$ relatively compact in $U$, and $f: U'\rightarrow U $ a $\mathbb{C}$-analytic mapping, proper of degree $d$. Let $L \subset U' $be a compact connect subset containing $f^{-1}\left(\overline{U'}\right)$ and the critical points of $f$, and such that $X_0=U-L$ is connected. Let $X_n$ be a covering space of $X_0$ of degree $d^n$, $\rho_n:X_{n+1}\rightarrow X_n$ and $\pi_n:X_n\rightarrow X_0$ be the projections and let $X$ be the disjoint union of the $X_n$. For each $n$ choose a lifting $\widetilde{f}_n\colon \pi_n^{-1}(U'-L)\rightarrow X_{n+1},$ of $f$. Then $T$ is the quotient of $X$ by the equivalence relation identifying $x$ to $\widetilde{f}_n(x)$ for all $x\in \pi_n^{-1}(U'-L)$ and all $n=0,1,2,\ldots$. The open set $T'$ is the union of the images of the $X_n, n=1,2,\ldots$, and $F:T'\rightarrow T$ is induced by the $\rho_n$.
Why $T$ is a Riemann surface and isomorphic to an annulus of finit modulus? Is there anything special about the $\pi_n,\rho_n$? What kind of background do I need?