Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do not see a simple argument for that. Any help is welcome.



Hyperbolic structures on infinite type surfaces
This is a classical theorem: hyperbolic Riemann surfaces of finite hyperbolic area
are compact surfaces with finitely many punctures. Tsuji (Theorem XI.12) credits this to Siegel (1945). The proof is a simple computation of the area of the fundamental polygon of the uniformizing Fuchsian group.


Tsuji, Potential theory in modern function theory, Maruzen Ltd, Tokyo 1959.
(There is an AMS reprint).


