Möbius Groups

The definition and properties of Möbius Transformations are discussed elsewhere and we will focus our study from group perspectives.

Proposition. The Möbius maps form a group $M$ under composition.

Proof.

0. Möbius maps are closed under composition.

1. The identity is

\[1_{\mathbb{C}_\infty}(z) = {1z + 0 \over 0z + 1}\]

2. The inverse of $f(z)$ is

\[f^{-1}(z) = {dz - b \over -cz + a}\]

3. Composition of functions is always associative.

Matrix Representation

Proposition. The map $\theta: \text{GL}_2(\mathbb{C}) \to M$ where

\[A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto f_{A} \quad \text{where} \quad f_{A}(z) = {az + b \over cz + d}\]

is a surjective group homomorphism.

Proof.

$\theta$ lands in $M$ as $A \in \text{GL}_2(\mathbb{C})$ gives $ad - bc \not= 0$, which also shows surjectivity.

From the composition formula and matrix multiplication,

\[\theta(A_2)\theta(A_1)(z) = \theta(A_2A_1)(z)\]

and hence $\theta$ is a surjective group homomorphism.

Proposition. The kernel of $\theta$ is

\[\ker \theta = \Set{A : z = {az + b \over cz + d}} = \Set{\lambda I : \lambda \in \mathbb{C}, \lambda \not= 0} = Z\]

Definition. The projective general linear group is defined by

\[\text{PGL}_2(\mathbb{C}) = \text{GL}_2(\mathbb{C}) / Z\]

By Isomorphism Theorem, $M \cong \text{PGL}_2(\mathbb{C})$.

Proposition. As $f_A = f_B$ iff $B = \lambda A$, we also have $\theta: \text{SL}_2(\mathbb{C}) \to M$ an surjective group homomorphism, with kernel $\Set{\pm I}$ and hence $M$ is isomorphic to the quotient group $\text{SL}_2/\Set{\pm I}$.

Definition. The projective special linear group is defined by

\[\text{PSL}_2(\mathbb{C}) = \text{SL}_2(\mathbb{C}) / \Set{\pm I}\]

By Isomorphism Theorem, $M \cong \text{PSL}_2(\mathbb{C})$.

Conjugacy

Proposition. Any Möbius map is conjugate to $f(z) = vz$ for some $v \not= 0$, or to $f(z) = z + 1$.

Proof.

From the above, $\theta: \text{GL}_2(\mathbb{C}) \to M$ is a surjective homomorphism.

The conjugacy classes in $\text{GL}_2(\mathbb{C})$ are

\[\begin{align*} \begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix} \quad &\longmapsto \quad g(z) = {\lambda z + 0 \over 0z + \mu} = {\lambda \over \mu}z \\ \\ \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} \quad &\longmapsto \quad g(z) = {\lambda z + 0 \over 0z + \lambda} = z \\ \\ \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} \quad &\longmapsto \quad g(z) = {\lambda z + 1 \over 0z + \lambda} = z + {1 \over \lambda} \\ \end{align*}\]

In fact, $\begin{pmatrix} 1 & {1 \over \lambda} \\ 0 & 1 \end{pmatrix}$ is conjugate to $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$:

\[\begin{pmatrix} \lambda & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & {1 \over \lambda} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} {1 \over \lambda} & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \quad \longmapsto \quad g(z) = z + 1\]

The property that can be transferred to conjugates is the number of fixed point. From the above, we can see that $g(z) = vz$ fixes $0$ and $\infty$ and $g(z) = z + 1$ fixes $\infty$ only. Therefore,

Proposition. Every non-identity Möbius map has exactly one or two fixed points.

Proof.

For any $g \in M, g \not= 1_M$, there is $h \in M$ such that $hgh^{-1}(z) = vz$ or $z + 1$.

Suppose $g$ fixes $w$, we have

\[g(w) = w \iff hgh^{-1}(h(w)) = h(w)\]

so $g$ and $hgh^{-1}$ has the same number of fixed points. Hence, $g$ has either exactly $2$ (for $hgh^{-1}(z) = vz$) or exactly $1$ (for $hgh^{-1}(z) = z + 1$) fixed points.

Finally, any Möbius map with at least three fixed points is the identity.

Proposition. Let $f$ be a Möbius map fixes $w$, then for any Möbius map $h$, the conjugate $hfh^{-1}$ fixes $h(w)$. If otherwise $hfh^{-1}$ fixes $w$, then $f$ fixes $h^{-1}(w)$.

Proof.

If $f(w) = w$, $hfh^{-1}(h(w)) = hf(w) = h(w)$.

If $hfh^{-1}(w) = w$, $f(h^{-1}(w)) = h^{-1}(w)$.

Acting on $C_\infty$

Definition. An action of $G$ on $X$ is called three-transitive if there is always a $g \in G$ such that for any two triples $x_1, x_2, x_3$ and $y_1, y_2, y_3$ of distinct elements of $X$, $g(x_i) = y_i$. If this $g$ is unique, the action is called sharply three-transitive.

Proposition. The Möbius group $M$ acts sharply three-transitively on $C_\infty$.

References