Kleinian and Fuchsian groups

Let be a group of Möbius transformations. We say that acts **properly discontinuously** at a point of
if
there exists an open neighbourhood of such that the subset of
defined by

With the topology of Möbius transformations explained above we have the following result.

**Remark**: frequently the term Kleinian is used for a discrete
groups of Möbius transformations, even if
(Thurston's
definition). This is motivated by the fact that a discrete group of Möbius
transformations acts properly discontinuously on the -dimension
hyperbolic space. See proposition 1.17 below for a similar
result for Fuchsian groups.

An interesting property of Kleinian groups is that their limit sets have at most two points or are uncountable (a similar property holds for iteration of rational functions on the Riemann sphere).

A Kleinian group whose limit set consists of at most two points is
called an **elementary** group.

Elementary groups appear as groups of deck transformations of branched
coverings of certain Riemann surfaces. We can list them all, but before
that we need to set up some notation. Let and be two
Riemann surfaces and
a mapping. We say that
is a **branched covering** if at every point of we have that
is either a covering in the usual sense (called regular covering
by some authors), or is locally like
for some
positive integer . In the latter case we say that is a
ramification point and is called the ramification value of .
We also require that the set of points with ramification is a discrete
subset of . If is a compact surface of genus , is
simply connected, and is ramified over
, with
ramification values
, we say that has
**signature**
. For notation purposes we
allow the symbol in a signature, to include punctures.
A **puncture** is just a missing

point of (or,
more formally, a domain on conformally equivalent to the unit disc
minus the origin). We include the punctures of a surface in its
signature with the symbol ; for example, the complex
plane is equal to the Riemann sphere with one puncture, so its signature
is
, and the complex plane minus the origin has
signature
. With this notation we can describe all
elementary groups.

Suppose is an elementary, non trivial group with no limit points. Then is the deck transformations of a covering , branched over two or three points. The possible signatures of such covering are:

If has one limit point then (after conjugation) acts properly on the complex plane; that is, it is the deck transformation group of a (possibly branched) covering of the form . We have the following possible signatures:

The first signature corresponds to a torus and the second one to the complex plane minus one point; the other signatures correspond to different branched coverings of the complex plane or the Riemann sphere.

If has two limit points we have that it acts (after conjugation) on ; the corresponding coverings are those of a torus (signature ) and a sphere with four ramification points of order (signature ).

A Kleinian group is called **Fuchsian** if it leaves a disc
invariant. Here by a disc (in the Riemann sphere) we understand some
circular disc or a half-plane. The boundary of is called the circle
at infinity of . We say that the group is of the **first kind** if
every point of the circle at infinity is a limit point (
);
otherwise the group is called of the **second kind** (
). By conjugation in
we can assume,
when necessary, that a Fuchsian group leaves the upper half plane
invariant; in that case can be considered as a subgroup of
.

We set the convention and for a signature we put (the negative of the Euler characteristic of the covering). Then we have that for an elementary group with no limit points, , while for an elementary group with one or two limit points we get . The next result says that if a signature satisfies then it is the signature of a (possibly branched) covering defined on the upper half plane.

Pablo Ares Gastesi 2005-08-31