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My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):

Proposition. The convex core of a complete hyperbolic surface has finite area if and only if the surface is of finite type.

Question. What is the best literature to cite for this result?

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Yes, this is a theorem of Siegel, and a reference with simple proof is Theorem XI.12 in M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo 1959 (there is an AMS Chelsea reprint, 1975).

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