Riemann Surface of an Annulus Could someone please show me how to prove this step by step? This is a Practice Question for the final exam but we barely touched on Riemann Surface in the lectures X.X'
Thanks! Especially how to construct charts and the biholomorphic functions.

Let $\lambda > 1$. Consider the Riemann surface defined as
$$X := \mathbb{H} \big / \sim$$
where $z \sim w$ if and only if $w = \lambda^n z$ for some $n \in \mathbb{Z}$.
Prove that $X$ is a Riemann surface, and show that it is isomorphic to the annulus
$$A_R := \{ z \in \mathbb{C} \ : \ 1 < |z| < R \}$$
for some value of $R$. Find $R$ as a function of $\lambda$.

 A: Interesting question!
It shouldn't be hard to see that $X$ is a Riemann surface, and one shouldn't need to resort to explicit holomorphic charts.  The idea is that $\mathbb{H}$ is a complex surface, and we are quotienting it by a map $z \mapsto \lambda z$ that preserves this complex structure.  In other words, we can regard sufficiently small neighborhoods of $\mathbb{H}$ as charts on $X$, and the transition functions will be of the form $z \mapsto \lambda^n z$ which is holomorphic.  
Now we want to classify the resulting Riemann surface $X$.  Generally, it's hard to obtain an explicit uniformization map, and I haven't thought enough to see whether it's doable in this case.  But one can proceed indirectly.  Firstly, observe that from the description $X = {\mathbb{H}/\sim}$ it is manifest that $\pi_1(X) \cong \mathbb{Z}$.  Any doubly-connected Riemann surface is isomorphic to either $\mathbb{C}^*$, $\Delta^*$, or an annulus $A_R$ for $1 < R < \infty$ (as defined in your question).  We can distinguish these by their modulus.  
In the metric $\rho = \lvert dz \rvert/\lvert z \rvert$, $X$ is presented as a right cylinder with circumference $\log \lambda$ and height $\pi$.  To see this, note that $\{z \in \mathbb{H} \mid 1 \leq \lvert z \rvert \leq \lambda\}$ is a fundamental domain for $X$.  The circumference is computed by the hyperbolic distance from $i$ to $\lambda i$ along the vertical geodesic, and the height can be computed as the length of the semicircular arc joining $-1$ and $1$, where the hyperbolic metric coincides with the Euclidean metric.  Anyway, we find that the modulus of $X$ is given by $\pi/\log \lambda$.  
On the other hand, the modulus of $A_R$ is $\frac{\log R}{2\pi}$.  So solving for $R$ we find that $R = e^{2\pi^2/\log \lambda}$.  
