Riemann surfaces that are not of finite type

Question

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.

Answer 1

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.

Answer 2

1. Classification. You do not specify what classification (what is your equivalence relation?) Topological classification is due to Kerekjarto. A reference is given in the answer of Richard Kent. Complete conformal classification is hopeless.

2. There are many different notions of hyperbolicity. They all coincide in the simply connected case.

a) Universal cover is the disc. (All Riemann surfaces, except tori, sphere, plane and cylinder are hyperbolic in this sense).

b) There is no Green function.

c) There is no positive harmonic function

d) There is no bounded analytic function

e) There is no analytic function with finite Dirichlet integral

f) And so on.

In 1950-s there was large area of research called Classification of Riemann surfaces. The main subject of this research was establishing the relations between b)-f) and other similar properties, and finding criteria for concrete surfaces to satisfy b)-e).

You can find a lot of results of this sort in the books of Tsuji, Potential theory in modern function theory, Nevanlinna, Uniformisation, and Ahlfors and Sario book on Riemann surfaces.

1. Hurwitz automorphism theorem is not true. There are many open surfaces with a rich group of automorphisms.

On your last remark. The subject is out of fasion, so modern books do not treat it. There are just too many Riemann surfaces to have an interesting classification of all of them. So the research is concentrated on various special classes that have some applications. For example, on hyperelliptic surfaces (those obtained as a 2-sheeted ramified covering of the plane); Such surfaces occur in mathematical physics.

Another interesting class which is studied is Abelian coverings of compact surfaces.