A Note on m-Möbius Transformations

Dorin Ghisa

York University, Toronto, Canada.

DOI: 10.4236/apm.2021.1111057                

Abstract

Lie groups of bi-Möbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-Möbius transformations are introduced and similar problems as those related to bi-Möbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-Möbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-Möbius transformations are also studied.

Keywords

Möbius Transformation, Complex Manifold, Lie Group

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 $f\mathrm{<mo>\; :\; </mo>}\overline{ℂ}\times \overline{ℂ}\to \overline{ℂ}$ of the form:

$f\left(z_1\mathrm{,}{z}_2\right)=\frac{A{z}_1{z}_2+a\left(1-z_1-z_2\right)}{a\left(z_1{z}_2-z_1-z_2\right)+A}$, where $a\in ℂ\backslash \left\{\mathrm{0,1}\right\}$ and $A=a^2-a+1$

Proposition 1: The function $f\left(z_1\mathrm{,}{z}_2\right)=z_1\circ z_2$ is a composition law in $\overline{ℂ}$ satisfying:

a) $z_1\circ z_2=z_2\circ z_1$ for every $z_1\mathrm{,}{z}_2\in \overline{ℂ}$

b) $z\circ 1=1\circ z=z$ for every $z\in \overline{ℂ}$

c) $z\circ \left(1/z\right)=1$ for every $z\in \overline{ℂ}$

d) $\left(1/z_1\right)\circ \left(1/z_2\right)=1/\left(z_1\circ z_2\right)$ for every $z_1\mathrm{,}{z}_2\in \overline{ℂ}$

e) $z_1\circ \left(z_2\circ z_3\right)=\left(z_1\circ z_2\right)\circ z_3$ for every $z_1\mathrm{,}{z}_2\mathrm{,}{z}_3\in \overline{ℂ}$

f) $z_1\circ z_2=a$ if and only if $z_1=a$ or $z_2=a$ and $z_1\circ z_2=1/a$ if and only if $z_1=1/a$ or $z_2=1/a$.

It is obvious that this composition law defines a structure of Abelian group on $\overline{ℂ}$ 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 $G_a$ of this group. Since $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ is a differentiable manifold on which the group operations are conformal mappings the subgroup $G_a$ is a Lie group.

Theorem 1. For every $z\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ the group $\langle z\rangle$ generated by z is a subgroup of $G_a$.

Proof: Let us denote $z^{\left(n+1\right)}=z\circ z^{\left(n\right)}$, every $n\in \mathbb{Z}$, where $z^{\left(0\right)}=1$ and $z^{\left(1\right)}=z$ and notice that $z\circ z^{\left(0\right)}=z\circ 1=z$. An easy induction argument shows that for every $m\mathrm{,}n\in \mathbb{Z}$ we have $z^{\left(m\right)}\circ z^{\left(n\right)}=z^{\left(m+n\right)}$ and in particular $z^{\left(n\right)}\circ z^{\left(-n\right)}=z^{\left(0\right)}=1$, which means that indeed $\langle z\rangle$ is a subgroup of $G_a$. Let us notice that, for $z\ne 1$ we have $z^{\left(n\right)}=1$ if and only if $n=0$.

If $z_1\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ then $g_{z_1}\left(z_2\right)=f\left(z_1,z_2\right)=\frac{\left(A{z}_1-a\right)z_2-a\left(z_1-1\right)}{a\left(z_1-1\right)z_2+A-a{z}_1}$ is a Möbius transformation in $z_2$ and if $z_2\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ then $h_{z_2}\left(z_1\right)=f\left(z_1,z_2\right)=\frac{\left(A{z}_2-a\right)z_1-a\left(z_2-1\right)}{a\left(z_2-1\right)z_1+A-a{z}_2}$ is a Möbius transformation in $z_1$. Indeed, $\left(A{z}_1-a\right)\left(A-a{z}_1\right)+a^2{\left(z_1-1\right)}^2=0$ if and only if $z_1=a$ or $z_1=1/a$, which has been excluded and similarly $\left(A{z}_2-a\right)\left(A-a{z}_2\right)+a^2{\left(z_2-1\right)}^2=0$ if and only if $z_2=a$ or $z_2=1/a$, which again has been excluded. These properties justify the name of bi-Möbius we have given to $f\left(z_1\mathrm{,}{z}_2\right)$.

Couples of bi-Möbius transformations generate mappings $M\mathrm{<mo>\; :\; </mo>}{\overline{ℂ}}^2\to {\overline{ℂ}}^2$ of the form $M\left(z_1\mathrm{,}{z}_2\right)=\left(f_1\left(z_1\mathrm{,}{z}_2\right)\mathrm{,}{f}_2\left(z_1\mathrm{,}{z}_2\right)\right)$, where

$f_k\left(z_1,z_2\right)=\frac{{\omega }_k{z}_1{z}_2-z_1-z_2+1}{z_1{z}_2-z_1-z_2+{\omega }_k}$, and ${\omega }_k=a_k+1/a_k-1$, $k=\mathrm{1,2}$. The Proposition 1, f) shows that such a mapping has a set E of four fixed points, namely

$\left(a_1\mathrm{,}{a}_2\right)$, $\left(1/a_1\mathrm{,}{a}_2\right)$, $\left(a_1\mathrm{,}1/a_2\right)$ and $\left(1/a_1\mathrm{,}1/a_2\right)$. When restricting M to ${\overline{ℂ}}^2\backslash E$ its components are bijective mappings in each one of the variables. Indeed, if $z_1\in \overline{ℂ}\backslash \left\{a_k\mathrm{,}1/a_k\right\}$, $k=\mathrm{1,2}$, then $f_k\left(z_1\mathrm{,}{z}_2\right)$ is a Möbius transformation in $z_2$, hence it is a bijective mapping of $\overline{ℂ}$ and since $f_k\left(z_1\mathrm{,}{a}_k\right)=a_k$ and $f_k\left(z_1\mathrm{,}1/a_k\right)=1/a_k$, it is a bijective mapping of $\overline{ℂ}\backslash \left\{a_k\mathrm{,}1/a_k\right\}$ onto itself. Similarly, if $z_2\in \overline{ℂ}\backslash \left\{a_k\mathrm{,}1/a_k\right\}$, $k=\mathrm{1,2}$, then $f_k\left(z_1\mathrm{,}{z}_2\right)$ is Möbius in $z_1$, hence it is a bijective mapping of $\overline{ℂ}\backslash \left\{a_k\mathrm{,}1/a_k\right\}$ onto itself. Since $f_k\left(z_1,z_2\right)=f_k\left(z_2,z_1\right)$ we have $M\left(z_1,z_2\right)=M\left(z_2,z_1\right)$ hence M is not injective. However, by factorizing ${\overline{ℂ}}^2$ with the two elements group $\langle \sigma \rangle$ generated by the symmetry $\sigma \left(z_1\mathrm{,}{z}_2\right)=\left(z_2\mathrm{,}{z}_1\right)$, M induces a bijective mapping of $\tilde{M}$ of ${\overline{ℂ}}^2\mathrm{<mo>\; /\; </mo>}\langle \sigma \rangle$ onto ${\overline{ℂ}}^2$. Indeed, an easy computation shows that for fixed ${\omega }_1$ and ${\omega }_2$ the equations $f_1\left(z_1\mathrm{,}{z}_2\right)=b_1$ and $f_2\left(z_1\mathrm{,}{z}_2\right)=b_2$ determine uniquily $z_1+z_2$ and $z_1{z}_2$ belonging to ${\overline{ℂ}}^2/\langle \sigma \rangle$. We can call this mapping Möbius transformation of ${\overline{ℂ}}^2/\langle \sigma \rangle$. This is a new concept. We are expecting Möbius transformations of ${\overline{ℂ}}^2/\langle \sigma \rangle$ to have similar properties with those of Möbius transformations of $\overline{ℂ}$, as well as lot of applications. Any such Möbius transformation depends on two complex parameters: ${\omega }_1$ and ${\omega }_2$. A composition law in the set of these transformations can be defined in the following way. Let:

$\begin{array}{l}{w}_1=\frac{\left({\omega }_1{z}_2-1\right)z_1-z_2+1}{\left(z_2-1\right)z_1-z_2+{\omega }_1},w_2=\frac{\left({\omega }_2{z}_2-1\right)z_1-z_2+1}{\left(z_2-1\right)z_1-z_2+{\omega }_2},\\ {\zeta }_1=\frac{\left({\omega }_3{w}_2-1\right)w_1-w_2+1}{\left(w_2-1\right)w_1-w_2+{\omega }_3},{\zeta }_2=\frac{\left({\omega }_4{w}_2-1\right)w_1-w_2+1}{\left(w_2-1\right)w_1-w_2+{\omega }_4}\end{array}$

Let us notice that since ${\zeta }_1$ is a Möbius transformation in $w_1$ for every $w_2\in \overline{ℂ}\backslash \left\{a_3\mathrm{,}1/a_3\right\}$ and $w_1$ is a Möbius transformation in $z_1$ for every $z_2\in \overline{ℂ}\backslash \left\{a_1\mathrm{,}1/a_1\right\}$, then ${\zeta }_1$ is a Möbius transformation in $z_1$ for every $w_2\in \overline{ℂ}\backslash \left\{a_3\mathrm{,}1/a_3\right\}$ and $z_2\in \overline{ℂ}\backslash \left\{a_1\mathrm{,}1/a_1\right\}$. Analogously it can be shown that ${\zeta }_1$ is a Möbius transformation in $z_2$ and that ${\zeta }_2$ is a Möbius transformation in $z_1$ and in $z_2$ when excluding some points, in other words $\left({\zeta }_1\mathrm{,}{\zeta }_2\right)=\left({\phi }_1\left(z_1\mathrm{,}{z}_2\right)\mathrm{,}{\phi }_2\left(z_1\mathrm{,}{z}_2\right)\right)$, where ${\phi }_k\left(z_1\mathrm{,}{z}_2\right)$ are Möbius transformations in $z_1$ when some values of $z_2$ are omitted and they are Möbius transformation in $z_2$ when some values of $z_1$ are omitted. Their expressions appear to be more complicated than those of $f_k\left(z_1\mathrm{,}{z}_2\right)$. However, they induce Möbius transformation of ${\overline{ℂ}}^2/\langle \sigma \rangle$.

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 $f\left(z_1\mathrm{,}f\left(z_2\mathrm{,}{z}_3\right)\right)$ is a Möbius transformation in each one of the variables as long as the other variables belong to $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$.

To simplify the writing, let us denote $\omega =a+1/a-1$, $s_2^{\left(2\right)}=z_1{z}_2$ and $s_1^{\left(2\right)}=z_1+z_2$, $s_3^{\left(3\right)}=z_1{z}_2{z}_3$, $s_2^{\left(3\right)}=z_1{z}_2+z_1{z}_3+z_2{z}_3$, $s_1^{\left(3\right)}=z_1+z_2+z_3$, $\cdots$, $s_n^{\left(n\right)}=z_1{z}_2\cdots z_n$, $\cdots$, $s_1^{\left(n\right)}=z_1+z_2+\cdots +z_n$. When no confusion is possible we can get rid of the upper subscript. Then, after a little calculation, we get:

$f_2\left(z_1,z_2\right)=f\left(z_1,z_2\right)=\frac{\omega s_2-s_1+1}{s_2-s_1+\omega }$

$f_3\left(z_1,z_2,z_3\right)=f\left(f_2\left(z_1,z_2\right),z_3\right)=\frac{\left(1+\omega \right)s_3-s_2+1}{s_3-s_1+\left(1+\omega \right)}$

$f_4\left(z_1,z_2,z_3,z_4\right)=f\left(f_3\left(z_1,z_2,z_3\right),z_4\right)=\frac{\left({\omega }^2+\omega -1\right)s_4-\omega s_3+s_2-s_1+\omega }{\omega s_4-s_3+s_2-\omega s_1+\left({\omega }^2+\omega -1\right)}$

A pattern appears regarding the coefficients of $s_k$ in these expressions, namely in every $f_m$ the coefficient of $s_k$ at the numerator is the same as the coefficient of $s_{m-k}$ 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 $f_m\left(z_1,z_2,\cdots ,z_m\right)=\frac{a_0{s}_n+a_1{s}_{n-1}+\cdots +a_m}{b_0{s}_n+b_1{s}_{n-1}+\cdots +b_m}$, then for every $k=1,2,\cdots ,m$ we have $b_k=a_{m-k}$.

The function $f_m\left(z_1,z_2,\cdots ,z_m\right)$ is a m-Möbius transformation, i.e. for every $k=1,2,\cdots ,m$ the function $f_m$ is a Möbius transformation in $z_k$ for any value of the other variables different of a and 1/a.

Proof: Let us denote ${\sigma }_k=\frac{1}{z_1{z}_2\cdots z_k}+\cdots +\frac{1}{z_{m-k+1}{z}_{m-k+2}\cdots z_m}$ for every $k=1,2,$ $\cdots ,m$ and suppose that $f_{m-1}\left(z_1,z_2,\cdots ,z_{m-1}\right)=1/f_{m-1}\left(1/z_1,1/z_2,\cdots ,1/z_{m-1}\right)$, which is obvious for $m=3,4,5$. We have:

$\begin{array}{c}{f}_m\left(z_1,z_2,\cdots ,z_m\right)=\frac{\left(\omega z_m-1\right)f_{m-1}\left(z_1,z_2,\cdots ,z_{m-1}\right)+\left(1-z_m\right)}{\left(z_m-1\right)f_{m-1}\left(z_1,z_2,\cdots ,z_{m-1}\right)+\left(\omega -z_m\right)}\\ =\frac{\left(\omega -1/z_m\right)\left[1/f_{m-1}\left(1/z_1,1/z_2,\cdots ,1/z_{m-1}\right)\right]+\left(1/z_m-1\right)}{\left(1-1/z_m\right)\left[1/f_{m-1}\left(1/z_1,1/z_2,\cdots ,1/z_{m-1}\right)\right]+\left(\omega /z_m-1\right)}\\ =\frac{\left(1/z_m-1\right)f_{m-1}\left(1/z_1,1/z_2,\cdots ,1/z_{m-1}\right)+\left(\omega -1/z_m\right)}{\left(\omega /z_m-1\right)f_{m-1}\left(1/z_1,1/z_2,\cdots ,1/z_{m-1}\right)+\left(1-1/z_m\right)}\\ =1/f_m\left(1/z_1,1/z_2,\cdots ,1/z_m\right)\end{array}$

If $f_m\left(z_1,z_2,\cdots ,z_m\right)=\frac{a_0{s}_n+a_1{s}_{n-1}+\cdots +a_m}{b_0{s}_n+b_1{s}_{n-1}+\cdots +b_m}$, then

$\begin{array}{c}{f}_m\left(1/z_1,1/z_2,\cdots ,1/z_m\right)=\frac{a_0{\sigma }_m+a_1{\sigma }_{m-1}+\cdots +a_m}{b_0{\sigma }_m+b_1{\sigma }_{m-1}+\cdots +b_m}=\frac{a_m{s}_n+\cdots +a_1{s}_1+a_0}{b_m{s}_m+\cdots +b_1{s}_1+b_0}\\ =1/f_m\left(z_1,z_2,\cdots ,z_m\right)=\frac{b_0{s}_m+b_1{s}_{m-1}+\cdots +b_m}{a_0{s}_m+a_1{s}_{m-1}+\cdots +a_m}\end{array}$

These last equalities are possible if and only if $b_k=a_{m-k}$. Simplifications may occur, as in the case of $f_5$ below, yet they do not alter the symmetry of the coefficients.

On the other hand, if we write

$f_m\left(z_1,z_2,\cdots ,z_m\right)=f\left(f_{m-1}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_{k-1}\mathrm{,}{z}_{k+1}\mathrm{,}\cdots \mathrm{,}{z}_{m-1}\right)\mathrm{,}{z}_k\right)$,

it is obvious that $f_m$ is a Möbius transformation in $z_k$ 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 $s_m$ are for a given m, we need to iteratively compute $f_j\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_j\right)$ for all the values of j from 2 to m. The expressions of these coefficients as functions of $\omega$ become more and more complicated. To illustrate this affirmation as well as the Theorem 1, let us notice that an elementary computation gives:

$f_5\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_5\right)=\frac{\omega \left(\omega +2\right)s_5-\left(\omega +1\right)s_4+s_3-s_1+\left(\omega +1\right)}{\left(\omega +1\right)s_5-s_4+s_2-\left(\omega +1\right)s_1+\omega \left(\omega +2\right)}$

$\begin{array}{l}{f}_6\left(z_1,z_2,\cdots ,z_6\right)\\ =\frac{\left({\omega }^3+2{\omega }^2-\omega -1\right)s_6-\left({\omega }^2+\omega -1\right)s_5+\omega s_4-s_3+s_2-\omega s_1+\left({\omega }^2+\omega -1\right)}{\left({\omega }^2+\omega -1\right)s_6-\omega s_5+s_4-s_3+\omega s_2-\left({\omega }^2+\omega -1\right)s_1+\left({\omega }^3+2{\omega }^2-\omega -1\right)}\end{array}$

$\begin{array}{l}{f}_7\left(z_1,z_2,\cdots ,z_7\right)\\ =\frac{\left({\omega }^3+3{\omega }^2+\omega -1\right)s_7-\omega \left(\omega +2\right)s_6+\left(\omega +1\right)s_5-s_4+s_2-\left(\omega +1\right)s_1+\omega \left(\omega +2\right)}{\omega \left(\omega +2\right)s_7-\left(\omega +1\right)s_6+s_5-s_3+\left(\omega +1\right)s_2-\omega \left(\omega +2\right)s_1+\left({\omega }^3+3{\omega }^2+\omega -1\right)}\end{array}$

$\begin{array}{l}{f}_8\left(z_1,z_2,\cdots ,z_8\right)\\ =\frac{\omega \left({\omega }^3+3{\omega }^2-3\right)s_8-\left({\omega }^3+2{\omega }^2-\omega -1\right)s_7+\left({\omega }^2+\omega -1\right)s_6-\omega s_5+s_4-s_3+\omega s_2-\left({\omega }^2+\omega -1\right)s_1+\left({\omega }^3+2{\omega }^2-\omega -1\right)}{\left({\omega }^3+2{\omega }^2-\omega -1\right)s_8-\left({\omega }^2+\omega -1\right)s_7+\omega s_6-s_5+s_4-\omega s_3+\left({\omega }^2+\omega -1\right)s_2-\left({\omega }^3+2{\omega }^2-\omega -1\right)s_1+\omega \left({\omega }^3+3{\omega }^2-3\right)}\end{array}$

3. Lie Groups of m-Möbius Transformations in $\overline{ℂ}$

For arbitrary z, $z_k\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$, $k=1,2,\cdots ,m$, let us denote $g_z=f_2\left(z\mathrm{,}{f}_m\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)\right)$, which is a set $G_m$ of m-Möbius transformations.

By Proposition 1 (see also [3] ), $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ endowed with the composition law $z\circ w=f_2\left(z\mathrm{,}w\right)$ is an Abelian group with the unit element 1 and for which the inverse element of z is $z^{-1}$. Moreover, an analytic atlas can be defined on $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ making it a differentiable manifold on which the group operations are conformal mappings and therefore this is a Lie group $G_a$. Basic knowledge about Lie groups can be found in [4]. A composition law in $G_m$ can be defined by $g_z\times g_w=g_{z\circ w}$. Then, for every z, $z_k\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$, $k=1,2,\cdots ,m$ we have $g_z\times g_1=g_{z\circ 1}=g_z$ and $g_z\times g_{z^{-1}}=g_{z\circ z^{-1}}=g_1$, hence $g_1$ is the unit element of this law and the inverse of $g_z$ is $g_{z^{-1}}\mathrm{.}$ Moreover, $g_z\times g_w=g_w\times g_z$.

Theorem 3. The set of m-Möbius transformations $G_m=\left\{g_z\mathrm{,}z\in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}\right\}$ with the composition law $g_z\times g_w=g_{z\circ w}$ is a Lie group.

Proof: Indeed, the properties we listed above show that $G_m$ is an Abelian group. It is isomorphic with $G_a$ under the mapping $\chi \left(z\right)=g_z$ since $\chi \left(z\circ w\right)=g_{z\circ w}=g_z\times g_w$ and $\chi \left(1\right)=g_1$. A topology on $G_m$ can be defined as the image by $\chi$ of the natural topology on $\overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$. This makes $G_m$ a differentiable manifold on which the composition law $g_z\times g_w=g_{z\circ w}$ 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 $\omega$ when indicating such a group.

Let $\zeta \in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ be arbitrary and for every $k\in \mathbb{Z}$ let us denote ${\zeta }^{\left(k+1\right)}=\zeta \circ {\zeta }^{\left(k\right)}$, where ${\zeta }^{\left(0\right)}=1$. It is obvious that for every $k\mathrm{,}l\in \mathbb{Z}$ we have ${\zeta }^{\left(k\right)}\circ {\zeta }^{\left(l\right)}={\zeta }^{\left(k+l\right)}$ and then $g_{{\zeta }^{\left(k\right)}}\times g_{{\zeta }^{\left(l\right)}}=g_{{\zeta }^{\left(k+l\right)}}$. In particular, $g_{{\zeta }^{\left(k\right)}}\times g_{{\zeta }^{\left(-k\right)}}$ $=g_{{\zeta }^{\left(0\right)}}=g_1$, hence the group $\langle g_{\zeta }\rangle$ generated by $g_{\zeta }$ is a subgroup of $G_m$.

Theorem 4. For every $\zeta \in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ the group $\langle g_{\zeta }\rangle$ is a discrete subgroup of $G_m$.

Proof: Indeed, if $\zeta =1$ then ${\zeta }^{\left(k\right)}=1$ for every $k\in \mathbb{Z}$. If $\zeta \ne 1$ then we have that ${\zeta }^{\left(k+1\right)}={\zeta }^{\left(k\right)}\circ \zeta \ne {\zeta }^{\left(k\right)}$. By using the expressions we have found for different $f_n$ we can easily check that there are values of $z\ne 1$ for which $f_n\left(z,z,\cdots ,z\right)=1$. For example, if $n=3$, $z^{\left(3\right)}=f_3\left(z,z,z\right)=1$ for every root of the equation $\omega z^2+\left(\omega -3\right)z+\omega =0$. Also, if $n=4$, then $z^{\left(4\right)}=1$ for every root of the equation $\left(\omega +1\right)z^4-4z^3+4z-\left(\omega +1\right)=0$ etc. It is obvious that for such values $\zeta =z$ the group $\langle \zeta \rangle$ is a cyclic one and so is the group $\langle g_{\zeta }\rangle$, hence it is a discrete subgroup of $G_n$.

If ${\zeta }^{\left(k\right)}\ne 1$ for every $k\ne 0$, then $\langle \zeta \rangle$ is not cyclic and ${\zeta }^{\left(k+1\right)}={\zeta }^{\left(k\right)}\circ \zeta \ne \zeta$ for every $k\ne 0$. Moreover, if $k\ne j$, then ${\zeta }^{\left(k-j\right)}={\zeta }^{\left(k\right)}\circ {\zeta }^{\left(-j\right)}\ne 1$, hence ${\zeta }^{\left(k\right)}\ne {\zeta }^{\left(j\right)}$. Suppose that there is a subsequence ( ${\zeta }^{\left(n_k\right)}$ ) of distinct elements such that ${\lim }_{n_k\to \infty }{\zeta }^{\left(n_k\right)}={\zeta }_0$. Let us split the sequence ( $n_k$ ) into two infinite subsequences ( $n_{k_1}$ ) and ( $n_{k_2}$ ) where $n_{k_1}+n_{k_2}=n_k$. Then ${\zeta }^{\left(n_k\right)}={\zeta }^{\left(n_{k_1}\right)}\circ {\zeta }^{\left(n_{k_2}\right)}$ and ${\zeta }_0={\lim }_{n_k\to \infty }{\zeta }^{\left(n_k\right)}={\lim }_{n_{k_1}\to \infty }{\zeta }^{\left(n_{k_1}\right)}\circ {\lim }_{n_{k_2}\to \infty }{\zeta }^{\left(n_{k_2}\right)}={\zeta }_0\circ {\zeta }_0$, which is possible if and only if ${\zeta }_0=1$, therefore ${\lim }_{n_k\to \infty }{\zeta }^{\left(n_k\right)}=1$. For every $j\in \mathbb{Z}$, ( ${\zeta }^{\left(n_k+j\right)}$ ) is a subsequence of ( ${\zeta }^{\left(k\right)}$ ) and ${\lim }_{n_k\to \infty }{\zeta }^{\left(n_k+j\right)}={\zeta }^{\left(j\right)}$, which again is possible only if ${\zeta }^{\left(j\right)}=1$. Yet ${\zeta }^{\left(j\right)}\ne 1$ if $j\ne 0$ and this shows that there is no convergent subsequence ( ${\zeta }^{\left(n_k\right)}$ ) of distinct elements. Hence the subgroup $\langle \zeta \rangle$ is discrete and so is $\langle g_{\zeta }\rangle$.

Corollary 1. For every $\zeta \in \overline{ℂ}\backslash \left\{a\mathrm{,}1/a\right\}$ the subgroup $\langle g_{\zeta }\rangle$ generated by $\zeta$ acts freely and properly discontinuously on $G_m$ by left and right translations.

4. Vector Valued m-Möbius Transformations

We can extend the concept of m-Möbius transformation to ${\overline{ℂ}}^n$ in the following way. For $a_k\in \overline{ℂ}\backslash \left\{\mathrm{0,1}\right\}$, let ${\omega }_k=a_k+1/a_k-1$, $k=1,2,\cdots ,n$, and let us build the m-Möbius transformations $f_{m\mathrm{,}k}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ as in Section 2 by using ${\omega }_k$ instead of $\omega$. We will study the function $f\mathrm{<mo>\; :\; </mo>}{\overline{ℂ}}^m\to {\overline{ℂ}}^n$ defined by

$f\left(z\right)=\left(f_{m\mathrm{,1}}\left(z\right)\mathrm{,}{f}_{m\mathrm{,2}}\left(z\right)\mathrm{,}\cdots \mathrm{,}{f}_{m\mathrm{,}n}\left(z\right)\right)$, where $z=\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$

Every $f_{m\mathrm{,}k}$ is a m-Möbius transformation of the form

$f_{m\mathrm{,}k}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)=\frac{a_0\left({\omega }_k\right)s_m+a_1\left({\omega }_k\right)s_{m-1}+\cdots +a_m\left({\omega }_k\right)}{a_m\left({\omega }_k\right)s_m+a_{m-1}\left({\omega }_k\right)s_{m-1}+\cdots +a_0\left({\omega }_k\right)}$, where $s_j$ are the

symmetric functions defined in Section 2, hence $f$ is a vector valued function whose every component is a m-Möbius transformation. For $w\in \Omega =\overline{ℂ}\backslash \left\{a_k\mathrm{,}1/a_k\mathrm{<mo>\; |\; </mo>}k=\mathrm{1,2,}\cdots \mathrm{,}n\right\}$ let

$g_w^{\left(k\right)}\left(z\right)=f_2\left(w\mathrm{,}{f}_{m\mathrm{,}k}\left(z\right)\right)=\frac{{\omega }_{k}w{f}_{m\mathrm{,}k}\left(z\right)-w-f_{m\mathrm{,}k}\left(z\right)+1}{w{f}_{m\mathrm{,}k}\left(z\right)-w-f_{m\mathrm{,}k}\left(z\right)+{\omega }_k}$, $k=1,2,\cdots ,n$. Then ${\Gamma }_{\Omega }=\left\{g_w\left(z\right)=\left(g_w^{\left(1\right)}\left(z\right)\mathrm{,}{g}_w^{\left(2\right)}\left(z\right)\mathrm{,}\cdots ,g_w^{\left(n\right)}\left(z\right)\right)\right\}$ is a set of vector valued functions whose components are all m-Möbius transformations.

Theorem 5. The composition law $g_z\cdot g_w=g_{z\circ w}$ induces a structure of Abelian group on ${\Gamma }_{\Omega }$ having the unit element $g_1$ and such that the inverse element of $g_z$ is $g_{z^{-1}}$.

Proof: Indeed, $g_z\cdot g_w=g_{z\circ w}=g_{w\circ z}=g_w\cdot g_z$, for every $z\mathrm{,}w\in \Omega$, $g_z\cdot g_1=g_{z\circ 1}=g_z$ for every $z\in \Omega$ and $g_z\cdot g_{z^{-1}}=g_{z\circ z^{-1}}=g_1$ for every $z\in \Omega$, since the same is true for every $g_z^{\left(k\right)}$ for every k, by Theorem 3, hence $g_{z^{-1}}=g_z^{-1}$.

Theorem 6. The mapping $c\mathrm{<mo>\; :\; </mo>}\Omega \to {\Gamma }_{\Omega }$ defined by $c\left(z\right)=g_z$ endows ${\Gamma }_{\Omega }$ with a Lie group structure.

Proof: The set ${\Gamma }_{\Omega }$ with the image topology induced by $c$ is a differentiable manifold and $c$ is a diffeomorphism. On the other hand, the group operations are conformal mappings and therefore of class $C^{\infty }$. Therefore the mapping $c$ is a Lie group isomorphism.

Let us notice that $f_{m\mathrm{,}k}\left(1\right)=f_{m\mathrm{,}k}\left(\mathrm{1,1,}\cdots \mathrm{,1}\right)=\frac{a_0\left({\omega }_k\right)C_m^m+a_1\left({\omega }_k\right)C_m^{m-1}+\cdots +a_m\left({\omega }_k\right)C_m^0}{a_m\left({\omega }_k\right)C_m^m+a_{m-1}\left({\omega }_k\right)C_m^{m-1}+\cdots +a_0\left({\omega }_k\right)C_m^0}=1$, hence $g_1^{\left(k\right)}\left(1\right)=f_2\left(\mathrm{1,}{f}_{m\mathrm{,}k}\left(1\right)\right)=1$, $k=\mathrm{1,2,}\cdots \mathrm{,}n$, hence $g_1\left(1\right)=1$.

When $m=n$ the function $f$ is a mapping of ${\overline{ℂ}}^m$ onto itself. It has a set E of $2^m$ fixed points. Indeed, every point $\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ where $z_k$ is either $a_k$ or $1/a_k$ is a fixed point of $f$.

The components of $f$ are m-Möbius transformations of $\overline{ℂ}$ in every variable $z_j$ if the other variables belong to $\Omega$.

Since, for fixed ${\omega }_k$, every $f_{m\mathrm{,}k}$ depends only on the symmetric sums $s_j$, the values of $f_{m\mathrm{,}k}\left(z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m\right)$ remain the same when making a permutation of the variables $z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m$. Therefore $f$ is not an injective function. Let ${\wp }_m$ be the group of permutations of $z_1\mathrm{,}{z}_2\mathrm{,}\cdots \mathrm{,}{z}_m$ and let ${\overline{ℂ}}^m/{\wp }_m$ be the factor space of ${\overline{ℂ}}^m$ with respect to this group. The function $f$ induces a bijective mapping $\tilde{f}$ of ${\overline{ℂ}}^m/{\wp }_m$ onto ${\overline{ℂ}}^m$. We can call it Möbius transformation of ${\overline{ℂ}}^m/{\wp }_m$. 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.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

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[2]	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. https://doi.org/10.1007/978-981-15-7775-8_33
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