I am interested in studying Riemann surfaces that are not of finite type. By a non-finite type Riemann surface, I mean a Riemann surface that is not conformally equivalent to any Riemann sub-surface of a compact Riemann surface. I have the following questions about such surfaces:-
[1]. Is there any classification theory for such surfaces like in the compact case?
[2]. Amongst such surfaces, what are the hyperbolic ones?
[3]. Under what additional conditions is something like the Hurwitz's automorphisms theorem true, if at all, for such surfaces?
Most of the books on Riemann surfaces I have looked through do not seem to treat such surfaces. Any reference to any book/paper that deals with such surfaces would be very helpful.
Best Answer
It should be pointed out that your definition of finite type is not the usual one. The usual definition is that a Riemann surface is of finite type if it is conformally equivalent to a compact Riemann surface minus a finite set of points. For instance, under this usual definition, an annulus of finite modulus is not of finite type. (Your definition also excludes things like the Riemann sphere minus a Cantor set.)
Donu Arapura is correct that the only exceptions to hyperbolicity are the usual ones, which answers 2.
As for 1, such things are classified topologically by the genus and the space of ends, by a theorem of I. Richards, see this. When you use the usual definition of finite type, then there is a Teichmuller theory for non-finite type surfaces with finitely generated fundamental group, where you specify boundary values when solving the Beltrami equation. You may read about this is most analytic treatments of Teichmuller theory, such as the books by Gardiner, Gardiner-Lakic, Nag, et cetera. I don't know about the theory in the infinitely generated case.
As for 3, any countable group is the automorphism group of some hyperbolic surface. The idea is that groups arise as automorphism groups of their Cayley graphs, and fattening the Cayley graphs into surfaces provides the result as long as you chose the lengths of the meridians carefully. This is a theorem of Allcock, see this.