Decomposition of hyperbolic surfaces near cusps into annuli 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.

May
23

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Decomposition of hyperbolic surfaces near cusps into annuli 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 $z\mapsto exp(2\pi i z)$.