##

Automorphisms of certain domains in the Riemann
sphere

Let
the set of complex numbers (complex plane) and
the **Riemann sphere**, the compactification of
by adding the point . We will use the following
notation for certain subsets of the complex plane:

A holomorphic mapping with a (necessarily holomorphic) inverse is called
biholomorphic. By an automorphism of an open set we understand a
biholomorphic mapping of the set onto itself.

The following results, describing the group of automorphisms of some
domains in the Riemann sphere are classical. But first we remark that
when we talk of mappings on
, or with values in the
Riemann sphere, we do not make any difference regarding the point of
, and thus we talk of holomorphic functions to include
meromorphic functions as well (think of holomorphic functions on Riemann
surfaces).

**Prop 1.1**
*The group of automorphisms of the Riemann sphere consists of
the group of Möbius transformations; that is, mappings of the form
, where , , and are complex
numbers satisfying
.*
Clearly we can multiply all coefficients of a Möbius transformation by a
non-zero number and the mapping does not change, so we can assume, if
needed, that
. Then the group of Möbius transformations is then
identified with the quotient of
by the
multiples of the identity, or with the quotient of
by . We denote these quotients by
and
,
respectively. We say that a matrix
represents a Möbius
transformation if is given by
.

**Prop 1.2**
*The group of holomorphic automorphisms of the complex plane
consists of the Möbius transformations of the form
,
where .*

**Prop 1.3**
*The group of holomorphic automorphisms of the upper half
plane consists of the Möbius transformations of the form
, with real coefficients satisfying
, or equivalently
.*

**Prop 1.4**
*The group of holomorphic automorphisms of the unit disc
consists of the Möbius transformations of the form
, where
and
are complex numbers.*
A more general statement is the following result of B. Maskit
[32], which we will not use but quote here for general
reference.

Pablo Ares Gastesi
2005-08-31