Contents

group theory

complex geometry

# Contents

## Idea

Let $k$ be a field, and let $\mathbb{P}^1(k)$ be the projective line over $k$. A Möbius transformation (also called a homography, a linear fractional transformation, or a fractional linear transformation which is my own preference – Todd) is a function $f: \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ defined by the rule

$f(x) = \frac{a x + b}{c x + d}$

where $a, b, c, d \in k$ and $a d - bc \in k^\times$. Möbius transformations form a group under composition, isomorphic to the projective linear group

$PGL_2(k) \coloneqq GL_2(k)/\{\lambda I: \lambda \in k^\times\}$

If, as in the case $k = \mathbb{C}$, each element of $k^\times$ has a square root, then this group is identified with

$PSL_2(k) \coloneqq SL_2(k)/\pm I.$

Alternatively, a fractional linear transformation can be considered as synonymous with an automorphism of the field of rational functions $k(x)$ as a field over $k$ of transcendence degree 1.

In complex analysis (which is the usual context when one speaks of Möbius transformations; otherwise one usually calls them by some combination of “linear” and “fractional”), Möbius transformations are precisely the biholomorphisms of the Riemann sphere, hence exactly its bijective conformal transformations.

Often, and particularly when $k$ is the the commutative ring of integers $\mathbb{Z}$, one considers a modular group where the coefficients $a, b, c, d$ are assumed to lie in an integral domain and $a d - b c = 1$. (The homotopy quotient of the upper half-plane by the group $PGL_2(\mathbb{Z})$ is the moduli stack of elliptic curves over the complex numbers.)

## Properties

### Cross-ratios

###### Proposition

The action $PGL_2(k) \times \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ is 3-transitive, i.e., any triplet of distinct points $(a, b, c)$ may be mapped to any other triplet of distinct points $(a', b', c')$ by applying a group element.

###### Proof

It suffices to consider $a' = 0, b' = 1, c' = \infty$ where one applies the transformation $x \mapsto \frac{(x-a)(b-c)}{(x-c)(b-a)}$.

This motivates the following definition: given a 4-tuple $(a, b, c, d)$ of distinct points in $\mathbb{P}^1(k)$, its cross-ratio is

$\gamma(a, b, c, d) = \frac{(d-a)(b-c)}{(d-c)(b-a)}.$

It is not hard to see that the group action preserves the cross-ratio, i.e., $\gamma(g \cdot a, g \cdot b, g \cdot c, g \cdot d) = \gamma(a, b, c, d)$. Moreover, the group action is transitive on each cross-ratio-equivalence class of 4-tuples.

In the case $k = \mathbb{C}$ where $\mathbb{P}^1$ is interpreted as the Riemann sphere, it turns out that the cross-ratio of a 4-tuple is a real number if and only if the four points lie on a circle (or a line which is a circle passing through $\infty$). Hence Möbius = conformal transformations take circles to circles.

### Action on hyperbolic space

As explained at Poincare group, the group $PSL_2(\mathbb{C})$ can be identified with those linear transformations of Minkowski space $\mathbb{R}^4$ that preserve the Minkowski form $Q$, are orientation-preserving, and take the forward light cone $\{v = (\vec{x}, t): Q(v) = 0, t \gt 0\}$ to itself. It follows that $PSL_2(\mathbb{C})$ acts on the hyperboloid sheet

$H^3 = \{v = (\vec{x}, t): Q(v) = 1, t \gt 0\}$

which is naturally identified with hyperbolic 3-space. There is a Poincaré disk model for $H^3$; consider the disk $D^3$ that is the intersection of the future cone $\{v = (\vec{x}, t): Q(v) \geq 0, t \gt 0\}$ with the hyperplane $t = 1$. Its interior is an open 3-disk $int(D^3)$ which can be placed in perspective with $H^3$ by considering lines through the origin in $\mathbb{R}^4$: each line that passes through a unique point in $H^3$ passes through a unique point of $int(D^3)$. In this way, $D^3$ is viewed as a natural compactification of $H^3$, and the action of $PSL_2(\mathbb{C})$ on $H^3$ induces an action of $PSL_2(\mathbb{C})$ on $D^3$. The restriction of this action to the boundary $S^2 = \partial D^3$ (“the heavenly sphere”) coincides with the action on the Riemann sphere $S^2 = \mathbb{P}^1(\mathbb{C})$.

## Modular group

The modular group $\Gamma$ is the subgroup $PSL_2(\mathbb{Z}) \hookrightarrow PSL_2(\mathbb{C})$ consisting of Möbius transformations $z \mapsto \frac{a z + b}{c z + d}$ where $a, b, c, d \in \mathbb{Z}$ and $a d - b c = 1$.

The group $PSL_2(\mathbb{R})$ acts on the upper half-plane $H = \{z \in \mathbb{C}: Im(z) \geq 0\}$ (or rather $H \cup \{\infty\}$ as a subspace of the Riemann sphere), by restriction of the action of $PSL_2(\mathbb{C})$ on the Riemann sphere. Indeed, the action of $PSL_2(\mathbb{R})$ takes the real line $\mathbb{R} \cup \{\infty\}$ to itself, and any element $f(z) = \frac{a z + b}{c z + d}$ takes $i$ to $(b + a i)(d - c i)/(c^2 + d^2)$, whose imaginary part $(ad - b c)/(c^2 + d^2) = 1/(c^2 + d^2)$ is positive. By continuity it follows that the action preserves the sign of the imaginary part, hence takes the upper-half plane $H$ to itself.

It is illuminating to think of complex numbers $\tau$ such that $Im(\tau) \gt 0$ as representing elliptic curves $E$. Indeed, the field of meromorphic functions on an elliptic curve (i.e., a complex projective curve $E$ of genus $1$, or a torus equipped with a structure of complex analytic 1-manifold) can be identified with a field of doubly periodic holomorphic functions $\mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ where $L = \langle \omega_1, \omega_2\rangle$ is a fundamental lattice (a discrete cocompact subgroup of the additive topological group $\mathbb{C}$) attached to $E$. In essence, this field is generated by Weierstrass elliptic functions $\wp(z), \wp'(z): \mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ (here $\wp'$ is the derivative of $\wp$) which satisfy a cubic algebraic relation

$(\wp')^2 = 4\wp^3 - g_2\wp - g_3$

where the constants $g_2, g_3$ are expressed as certain Eisenstein series in the fundamental periods $\omega_1, \omega_2$. The $\mathbb{Z}$-linear basis elements $\omega_1, \omega_2$ of the lattice may be arranged so that $\tau = \omega_2/\omega_1$ has positive imaginary part. Of course, if there is a homothety $z \mapsto \lambda z$ that takes a lattice $L$ to a lattice $L'$, then the elliptic curves $E = \mathbb{C}/L$ and $E' = \mathbb{C}/L'$ are analytically isomorphic, so the map

$\tau \mapsto \mathbb{C}/\langle 1, \tau\rangle$

gives a surjection from complex numbers with positive imaginary part to isomorphism classes of smooth elliptic curves. Thus we may restrict attention to lattices of the form $L = \langle 1, \tau \rangle$.

Of course, $L$ admits more than one such basis $(1, \tau)$, but for any other $(1, \tau')$ there is a linear transformation $\gamma \in \Gamma \coloneqq PSL_2(\mathbb{Z})$ such that $\gamma(\tau) = \tau'$. In summary, the orbit space

$\{\tau \in \mathbb{C}: Im(\tau) \gt 0\}/\Gamma$

is a coarse moduli space for elliptic curves. In this context, one often says that elliptic curves are paramatrized by the $j$-invariant, a certain modular form $j(\tau)$ defined on the upper half-plane such that $j(\tau) = j(\tau')$ if and only if $\tau' = \gamma \cdot \tau$ for some $\gamma \in \Gamma$.

Of course, in some cases there may be more than one $\gamma \in \Gamma$ that fixes a given $\tau$; this is notably the case when $\tau$ is a fourth root of unity or a sixth root of unity. A more refined approach then is to consider, instead of the coarse orbit space, the (compactified) moduli stack $(H \cup \{\infty\})//\Gamma$ for elliptic curves, as a central geometric object of interest.

### Abstract structure of modular group

As an abstract group, $\Gamma = PSL_2(\mathbb{Z})$ is a free product $\mathbb{Z}/(2) \ast \mathbb{Z}/(3)$; explicitly, we may take the generator of order $2$ to be given by the Moebius transformation $z \mapsto -1/z$, and the generator of order $3$ to be given by $z \mapsto (z-1)/z$.

This concrete group and certain of its subgroups, such as congruence subgroups, are fairly ubiquitous; for example the modular group appears in the theory of rational tangles and of trivalent maps, and these groups frequently crop up in the theory of buildings (stuff on hyperbolic buildings to be filled in).

It is also worth pointing out that $\Gamma$ is a quotient of the braid group $B_3$. Consider the standard Artin presentation of $B_3$, with two generators $\sigma_1$, $\sigma_2$ subject to the relation

$\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2.$

Then $z \coloneqq (\sigma_1 \sigma_2)^3$ is a central element of $B_3$, and there is a central extension

$1 \to \mathbb{Z} \stackrel{1 \mapsto z}{\to} B_3 \stackrel{q}{\to} \Gamma \to 1$

where $q$ is the unique homomorphism mapping $\sigma_1\sigma_2$ to $\lambda z. (z-1)/z$, and $\sigma_1\sigma_2\sigma_1$ to $\lambda z. \frac{-1}{z}$.

Named after August Möbius.