Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near this cusp the surface decomposes into horizontal annuli. My question is how one explicitely finds these annuli near a given cusp to compute their modulus. You can assume as in the mentioned paper that the cuvr has genus $0$. 



Decomposition of hyperbolic surfaces near cusps into annuli
There is an explicit geometric model for cusps in $-1$ curvature, which is obtained by conjugating the parabolic element associated to the cusp to $z\mapsto z+1$.


Cusps are isometric to $C\_\alpha = \{z\in {\bf H} \mid Im(z) > \alpha\} / <z\mapsto z+1>$ for some $\alpha$ that can be expressed as a function of the hyperbolic area of the cusp. 
$$
area(C\_\alpha) = \int\_\alpha^\infty \int\_0^1 {dxdy\over y^2} = 1/\alpha.
$$
The boundary of such model is a closed horocycle (the projection of the horizontal line $\{Im(z)=\alpha\}$ in $\bf H$). I guess that the annuli you are refering to are simply cylinders in the cusp bounded by closed horocycles, e.g. 
$$A\_n = \{ z\in {\bf H} \mid n < Im(z)\leq n+1\}/<z \mapsto z+1>.$$


