Originally published at 狗和留美者不得入内. You can comment here or there.
There is a theorem in Chapter 4 Section 5 of Schlag’s complex analysis text. I went through it a month ago, but only half understood it, and it is my hope that passing through it again, this time with writeup, will finally shed light, after having studied in detail some typical examples of such Riemann surfaces, especially tori, the conformal equivalence classes of which can be represented by the fundamental region of the modular group, which arise from quotienting out by lattices on the complex plane, as well as Fuchsian groups.
In the text, the theorem is stated as follows.
Theorem 4.12. Let and
with the property that
for all
,
- for all
, all fixed points of
in
lie outside of
,
- for all
compact, the cardinality of
is finite.
Under these assumptions, the natural projection is a covering map which turns
canonically onto a Riemann surface.
The properties essentially say that the we have a Fuchsian group acting on
without fixed points, excepting the identity. To show that quotient space is a Riemann surface, we need to construct charts. For this, notice that without fixed points, there is for all
, a small pre-compact open neighborhood of
denoted by
, so that
.
So, in no two elements are twice represented, which mean the projection
is the identity, and therefore we can use the
s as charts. The
s as Mobius transformations are open maps which take the
s to open sets. In other words,
with pairwise disjoint open sets
. From this, the
s are open sets in the quotient topology. In this scheme, the
s are the transition maps.
Finally, we verify that this topology is Hausdorff. Suppose and define for all
,
where is sufficiently small. Define
and suppose that
for all
. Then for some
and
we have
.
Since has finite cardinality, there are only finitely many possibilities for
and one of them therefore occurs infinitely often. Pass to the limit
and we have
or
, a contradiction.