1. Introduction
When investigating Lie groups of Möbius transformations of the Riemann sphere, we were brought in [1] [2] and [3] to the study of some bi-Möbius transformations. These are functions
of the form:
, where
and
Proposition 1: The function
is a composition law in
satisfying:
a)
for every
b)
for every
c)
for every
d)
for every
e)
for every
f)
if and only if
or
and
if and only if
or
.
It is obvious that this composition law defines a structure of Abelian group on
whose unit element is 1 and the inverse of any z is 1/z. By removing the elements a and 1/a we get a subgroup
of this group. Since
is a differentiable manifold on which the group operations are conformal mappings the subgroup
is a Lie group.
Theorem 1. For every
the group
generated by z is a subgroup of
.
Proof: Let us denote
, every
, where
and
and notice that
. An easy induction argument shows that for every
we have
and in particular
, which means that indeed
is a subgroup of
. Let us notice that, for
we have
if and only if
.
If
then
is a Möbius transformation in
and if
then
is a Möbius transformation in
. Indeed,
if and only if
or
, which has been excluded and similarly
if and only if
or
, which again has been excluded. These properties justify the name of bi-Möbius we have given to
.
Couples of bi-Möbius transformations generate mappings
of the form
, where
, and
,
. The Proposition 1, f) shows that such a mapping has a set E of four fixed points, namely
,
,
and
. When restricting M to
its components are bijective mappings in each one of the variables. Indeed, if
,
, then
is a Möbius transformation in
, hence it is a bijective mapping of
and since
and
, it is a bijective mapping of
onto itself. Similarly, if
,
, then
is Möbius in
, hence it is a bijective mapping of
onto itself. Since
we have
hence M is not injective. However, by factorizing
with the two elements group
generated by the symmetry
, M induces a bijective mapping of
of
onto
. Indeed, an easy computation shows that for fixed
and
the equations
and
determine uniquily
and
belonging to
. We can call this mapping Möbius transformation of
. This is a new concept. We are expecting Möbius transformations of
to have similar properties with those of Möbius transformations of
, as well as lot of applications. Any such Möbius transformation depends on two complex parameters:
and
. A composition law in the set of these transformations can be defined in the following way. Let:
Let us notice that since
is a Möbius transformation in
for every
and
is a Möbius transformation in
for every
, then
is a Möbius transformation in
for every
and
. Analogously it can be shown that
is a Möbius transformation in
and that
is a Möbius transformation in
and in
when excluding some points, in other words
, where
are Möbius transformations in
when some values of
are omitted and they are Möbius transformation in
when some values of
are omitted. Their expressions appear to be more complicated than those of
. However, they induce Möbius transformation of
.
The study of these mappings is worthwhile, yet it exceeds the purpose of this note.
2. Multi-Möbius Transformations
The properties e) and f) from Proposition 1 show that
is a Möbius transformation in each one of the variables as long as the other variables belong to
.
To simplify the writing, let us denote
,
and
,
,
,
,
,
,
,
. When no confusion is possible we can get rid of the upper subscript. Then, after a little calculation, we get:
A pattern appears regarding the coefficients of
in these expressions, namely in every
the coefficient of
at the numerator is the same as the coefficient of
at the denominator. It is reasonable to believe that this happens due to the properties a), d) and e) listed above. Indeed, we can prove:
Theorem 2. If
, then for every
we have
.
The function
is a m-Möbius transformation, i.e. for every
the function
is a Möbius transformation in
for any value of the other variables different of a and 1/a.
Proof: Let us denote
for every
and suppose that
, which is obvious for
. We have:
If
, then
These last equalities are possible if and only if
. Simplifications may occur, as in the case of
below, yet they do not alter the symmetry of the coefficients.
On the other hand, if we write
,
it is obvious that
is a Möbius transformation in
as long as the other variables do not take the values a and 1/a.
We notice that in order to find exactly what the coefficients of
are for a given m, we need to iteratively compute
for all the values of j from 2 to m. The expressions of these coefficients as functions of
become more and more complicated. To illustrate this affirmation as well as the Theorem 1, let us notice that an elementary computation gives:
3. Lie Groups of m-Möbius Transformations in
For arbitrary z,
,
, let us denote
, which is a set
of m-Möbius transformations.
By Proposition 1 (see also [3] ),
endowed with the composition law
is an Abelian group with the unit element 1 and for which the inverse element of z is
. Moreover, an analytic atlas can be defined on
making it a differentiable manifold on which the group operations are conformal mappings and therefore this is a Lie group
. Basic knowledge about Lie groups can be found in [4]. A composition law in
can be defined by
. Then, for every z,
,
we have
and
, hence
is the unit element of this law and the inverse of
is
Moreover,
.
Theorem 3. The set of m-Möbius transformations
with the composition law
is a Lie group.
Proof: Indeed, the properties we listed above show that
is an Abelian group. It is isomorphic with
under the mapping
since
and
. A topology on
can be defined as the image by
of the natural topology on
. This makes
a differentiable manifold on which the composition law
defines a structure of Lie group. Different complex numbers a define different Lie groups of m-Möbius transformations, yet all of these groups are obviously isomorphic, and therefore there is no need to specify the numbers a, or
when indicating such a group.
Let
be arbitrary and for every
let us denote
, where
. It is obvious that for every
we have
and then
. In particular,
, hence the group
generated by
is a subgroup of
.
Theorem 4. For every
the group
is a discrete subgroup of
.
Proof: Indeed, if
then
for every
. If
then we have that
. By using the expressions we have found for different
we can easily check that there are values of
for which
. For example, if
,
for every root of the equation
. Also, if
, then
for every root of the equation
etc. It is obvious that for such values
the group
is a cyclic one and so is the group
, hence it is a discrete subgroup of
.
If
for every
, then
is not cyclic and
for every
. Moreover, if
, then
, hence
. Suppose that there is a subsequence (
) of distinct elements such that
. Let us split the sequence (
) into two infinite subsequences (
) and (
) where
. Then
and
, which is possible if and only if
, therefore
. For every
, (
) is a subsequence of (
) and
, which again is possible only if
. Yet
if
and this shows that there is no convergent subsequence (
) of distinct elements. Hence the subgroup
is discrete and so is
.
Corollary 1. For every
the subgroup
generated by
acts freely and properly discontinuously on
by left and right translations.
4. Vector Valued m-Möbius Transformations
We can extend the concept of m-Möbius transformation to
in the following way. For
, let
,
, and let us build the m-Möbius transformations
as in Section 2 by using
instead of
. We will study the function
defined by
, where
Every
is a m-Möbius transformation of the form
, where
are the
symmetric functions defined in Section 2, hence
is a vector valued function whose every component is a m-Möbius transformation. For
let
,
. Then
is a set of vector valued functions whose components are all m-Möbius transformations.
Theorem 5. The composition law
induces a structure of Abelian group on
having the unit element
and such that the inverse element of
is
.
Proof: Indeed,
, for every
,
for every
and
for every
, since the same is true for every
for every k, by Theorem 3, hence
.
Theorem 6. The mapping
defined by
endows
with a Lie group structure.
Proof: The set
with the image topology induced by
is a differentiable manifold and
is a diffeomorphism. On the other hand, the group operations are conformal mappings and therefore of class
. Therefore the mapping
is a Lie group isomorphism.
Let us notice that
, hence
,
, hence
.
When
the function
is a mapping of
onto itself. It has a set E of
fixed points. Indeed, every point
where
is either
or
is a fixed point of
.
The components of
are m-Möbius transformations of
in every variable
if the other variables belong to
.
Since, for fixed
, every
depends only on the symmetric sums
, the values of
remain the same when making a permutation of the variables
. Therefore
is not an injective function. Let
be the group of permutations of
and let
be the factor space of
with respect to this group. The function
induces a bijective mapping
of
onto
. We can call it Möbius transformation of
. A lot of questions remain to be answered about these transformations.
5. Conclusions
To emphasize the importance of the topic we dealt with in this paper, let us present a citation from [5]: “Although more than 150 years have passed since August Ferdinand Möbius first studied the transformations that now bear his name, it is fair to say that the rich vein of knowledge which he hereby exposed is still far from being exhausted”.
The Möbius transformations are a chapter in any book of complex analysis. They have remarkable geometric properties and a lot of applications. The whole theory of automorphic functions is based on these transformations and they have surprising connections with the relativity theory. The concept of multi-Möbius transformation appears for the first time here and is related to the theory of Lie groups, which has itself deep connections with the Physics.
Acknowledgements
We thank Aneta Costin for her support with technical matters.