When investigating Lie groups of Möbius transformations of the Riemann sphere, we were brought in   and  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  ), 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 . 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
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.
To emphasize the importance of the topic we dealt with in this paper, let us present a citation from : “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.
We thank Aneta Costin for her support with technical matters.
 Barza, I. and Ghisa, D. (1997) Lie Groups Actions on the Möbius Strip, Topics in Complex Analysis. In: Dimiev, S. and Sekigawa, K., Eds., Differential Geometry and Physics, World Sci, Singapore, 62-72.
 Barza I. and Ghisa, D. (2020) Lie Groups Actions on Non Orientable Klein Surfaces. In: Dobrev, V., Ed., Lie Theory and Its Applications in Physics, Springer, Singapore, Vol. 335, 421-428.