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.



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Reference request: geometric finiteness of Fuchsian groups
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). 


