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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 $0$.

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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. $$A_n = \{ z\in {\bf H} \mid n < Im(z)\leq n+1\}/<z \mapsto z+1>.$$

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  • $\begingroup$ Thank for your answer. Is this easy in this case to compute the modulus of each annulus? $\endgroup$ May 23, 2015 at 18:56
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    $\begingroup$ Yes. The modulus of the annulus $\{\alpha < Im(z) < \beta \}/<z\mapsto z+1>$ is equal to $\beta - \alpha$. Note that this annulus is isomorphic to $\{z \in {\bf D} \mid exp(-2\pi\beta) < |z| < exp(-2\pi\alpha)\}$ through the map $z\mapsto exp(2\pi i z)$. $\endgroup$
    – coudy
    May 23, 2015 at 19:24
  • $\begingroup$ Thank you again. Is there a reference in which this relation between cusps and annuli etc. is nicely explained? Also, given a concrete example and a cusp, how can $\alpha$ in $C_{\alpha}$ be determined? $\endgroup$ May 25, 2015 at 9:30
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    $\begingroup$ Since you are interested by Teichmuller theory, I would suggest Hubbard, "Teichmuller theory". You will find there many results on the decomposition of a surface into cusps and pair of pants. $\endgroup$
    – coudy
    May 25, 2015 at 15:47

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