26 - Möbius Maps

Möbius Maps

Each element $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{GL}}_2(\mathbb{C})$ gives Möbius Map ${\Psi }_g:{\mathbb{C}}_{\infty }\to C_{\infty }$ given by

$${\Psi }_g(z):=\frac{az+b}{cz+d}$$

Need to be careful about $\infty$ so

• If $c\ne 0$ then define ${\Psi }_g\left(-\frac{d}{c}\right)=\infty$ and ${\Psi }_g(\infty )=\frac{a}{c}$

• If $c=0$ then define ${\Psi }_g(\infty )=\infty$

General Linear Group ${\mathrm{GL}}_2(\mathbb{C})$

General Linear Group ${\mathrm{GL}}_2(\mathbb{C})$ consists of $2\times 2$ matrices

$$g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\text{ with }a,b,c,d\in \mathbb{C}$$

Scalar-Uniqueness of the Möbius Maps

If $g,g^{\prime }\in {\mathrm{GL}}_2(\mathbb{C})$ give the same Möbius Map then

$$g=\lambda g^{\prime }\text{ for some }\lambda \ne 0$$

Composition of Möbius Maps

For $g_1,g_2\in {\mathrm{GL}}_2(\mathbb{C})$ then

$${\Psi }_{g_1{g}_2}={\Psi }_{g_1}\circ {\Psi }_{g_2}$$

so we have ${\mathrm{GL}}_2(\mathbb{C})$ acts on ${\mathbb{C}}_{\infty }$ via Möbius Maps

Translation Möbius Maps

Translation $z\mapsto z+a$ is Möbius Map ${\Psi }_{T(a)}$ where

$$T(a)=\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)$$

Dilation Möbius Maps

Dilation $z\mapsto bz$ is Möbius Map ${\Psi }_{D(b)}$ where

$$D(b)=\left(\begin{array}{cc}b& 0\\ 0& 1\end{array}\right)$$

Inversion Möbius Maps

Inversion $z\mapsto \frac{1}{z}$ is Möbius Map ${\Psi }_J$ where

$$J=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

Decomposition of a Möbius Map

Every Möbius Maps can be written as a composition of translation, dilations and inversions

Proof

Let ${\Psi }_g,g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ be the Möbius Map

Suppose first that $c\ne 0$ then without worrying about $\infty$ then

$$z\stackrel{{\Psi }_{D(c)}}{\to }cz\mapsto \stackrel{{\Psi }_{T(d)}}{\to }\mapsto cz+d\stackrel{{\Psi }_J}{\to }\frac{1}{cz+d}\stackrel{{\Psi }_{D\left(\frac{bc-ad}{c}\right)}}{\to }\frac{b-\frac{ad}{c}}{cz+d}\stackrel{{\Psi }_{T\left(\frac{a}{c}\right)}}{\to }\frac{az+b}{cz+d}$$

which is a chain of compositions

Hence we get

$${\Psi }_g={\Psi }_{T\left(\frac{a}{c}\right)}\circ {\Psi }_{D\left(\frac{bc-ad}{c}\right)}\circ {\Psi }_J\circ {\Psi }_{T(d)}\circ {\Psi }_{D(c)}$$

And also

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=T\left(\frac{a}{c}\right)\cdot D\left(\frac{bc-ad}{c}\right)\cdot J\cdot T(d)\cdot D(c)$$

Circline

Either

1. Circle in $\mathbb{C}$ (considered as a subset of $C_{\infty }$
2. Line in $\mathbb{C}$ (considered as a subset of ${\mathbb{C}}_{\infty }$) together with point $\{\infty \}$

Note that lines in $\mathbb{C}$ are given by equations of the form $|z-a|=|z-b|$ for $a$ and $b$ distinct in $\mathbb{C}$

Invariance of Circlines under Möbius Map

Möbius Map take circlines to circlines

Proof - Mini

By Decomposition of a möbius map,
It is enough to check this for translations, dilations and inversion
Hence is a straightforward case-by-case analysis