Möbius (linear-fractional) transformations

Intuition [Beginner]

A Möbius transformation is a particular kind of recipe for moving points of the plane around. You take four numbers $a,b,c,d$ and send each input $z$ to $(az+b)/(cz+d)$. The numbers must be chosen so that $ad-bc$ is not zero — otherwise the recipe collapses and gives the same value everywhere.

Three very simple moves are already Möbius transformations: shifting the plane by a fixed amount (translation, $z\mapsto z+b$), zooming in or out and rotating from the origin (scaling by a complex number, $z\mapsto az$), and turning the plane inside-out by swapping near and far through the origin (inversion, $z\mapsto 1/z$). What makes the family interesting is that every other Möbius transformation is built from these three by composition. The fancy formula $(az+b)/(cz+d)$ is not really new behaviour; it is bookkeeping that combines a translate, a scale, and an inversion into a single step.

These transformations are the natural symmetries of the [Riemann sphere]06.01.07. They map the sphere to itself, they send circles and straight lines to circles and straight lines, and they preserve angles between curves. That last property has a name — conformal — and it is the reason Möbius transformations show up everywhere conformal mapping matters.

Visual [Beginner]

The Möbius transformation $z\mapsto 1/z$ on the unit grid: horizontal and vertical lines bend into circles, the unit circle stays a circle, and angles between any pair of curves stay the same.

Worked example [Beginner]

Find the Möbius transformation that sends $0\mapsto -1$, $1\mapsto 0$, and $\infty \mapsto 1$.

The standard recipe is to use the cross-ratio. The function $z\mapsto (z-1)/(z+1)$ sends $1\mapsto 0$ (the numerator vanishes), $-1\mapsto \infty$ (the denominator vanishes), and $\infty \mapsto 1$ (both numerator and denominator grow at the same rate, and the leading coefficients are both $1$). So if we apply $z\mapsto (z-1)/(z+1)$ to the second triple $(-1,0,1)$ we get $(\infty ,-1,0)$, which is the first triple in reverse.

To go directly $0\mapsto -1$, $1\mapsto 0$, $\infty \mapsto 1$, write the unknown as $\varphi (z)=(az+b)/(cz+d)$ and read off equations from the three required outputs:

$$\varphi (0)=b/d=-1,\varphi (1)=(a+b)/(c+d)=0,\varphi (\infty )=a/c=1.$$

The first equation gives $b=-d$. The third gives $a=c$. The middle equation forces the numerator to vanish, so $a+b=0$, hence $b=-a$, hence $d=a$. Set $a=c=d=1$ and $b=-1$. The transformation is

$$\varphi (z)=\frac{z-1}{z+1}.$$

Check: $\varphi (0)=-1/1=-1$, $\varphi (1)=0/2=0$, $\varphi (\infty )=1/1=1$. All three outputs match.

What this tells us: three input-output pairs determine a unique Möbius transformation, and you can always solve for it by reading off three equations on $a,b,c,d$. The four numbers carry one redundancy (you can multiply them all by the same nonzero constant), so three equations pin down the answer.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let ${\mathbb{C}}_{\infty }:=\mathbb{C}\cup \{\infty \}$ denote the [Riemann sphere]06.01.07. A Möbius transformation (also called a linear-fractional or homographic transformation) is a map

$$\varphi :{\mathbb{C}}_{\infty }\to {\mathbb{C}}_{\infty },\varphi (z)=\frac{az+b}{cz+d},a,b,c,d\in \mathbb{C},ad-bc\ne 0,$$

with the conventions

$$\varphi (\infty )=\{\begin{array}{ll}a/c& c\ne 0,\\ \infty & c=0,\end{array}\varphi (-d/c)=\infty \text{ if }c\ne 0.$$

The non-vanishing of the determinant $\Delta :=ad-bc$ ensures $\varphi$ is non-constant: if $\Delta =0$, the rows $(a,b)$ and $(c,d)$ are proportional and the formula collapses to a constant. With $\Delta \ne 0$, $\varphi$ is a bijection of ${\mathbb{C}}_{\infty }$ onto itself, holomorphic with respect to the Riemann-surface structure on the sphere. ^{[Ahlfors Ch. 3 §3]}

Group structure. The composition of two Möbius transformations is again Möbius, with parameters given by matrix multiplication: if ${\varphi }_1$ has parameter matrix $M_1=\left(\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right)$ and ${\varphi }_2$ has $M_2$, then ${\varphi }_1\circ {\varphi }_2$ has parameter matrix $M_1{M}_2$. Two matrices $M,M^{\prime }$ define the same Möbius transformation iff $M^{\prime }=\lambda M$ for some $\lambda \in {\mathbb{C}}^{\ast }$ (multiplying $a,b,c,d$ by a common scalar leaves $(az+b)/(cz+d)$ unchanged). The group of Möbius transformations is therefore

$$\mathrm{M}\ddot{\mathrm{o}}\mathrm{b}({\mathbb{C}}_{\infty })={\mathrm{PGL}}_2(\mathbb{C}):={\mathrm{GL}}_2(\mathbb{C})/{\mathbb{C}}^{\ast }\cong {\mathrm{PSL}}_2(\mathbb{C}):={\mathrm{SL}}_2(\mathbb{C})/\{\pm I\},$$

where the second identification uses the fact that every class in ${\mathrm{PGL}}_2(\mathbb{C})$ has a representative of determinant $1$ (rescale $M$ by $1/\sqrt{\Delta }$), and the only such representatives differing by a common scalar are $M$ and $-M$.

Generation. Every Möbius transformation factors as a composition of three elementary types:

• Translations $z\mapsto z+\beta$ (with $\beta \in \mathbb{C}$),
• Scalings $z\mapsto \alpha z$ (with $\alpha \in {\mathbb{C}}^{\ast }$, encompassing both rotation by $\arg \alpha$ and dilation by $|\alpha |$),
• Inversion $z\mapsto 1/z$.

If $c=0$, then $\varphi (z)=(a/d)z+(b/d)$, a scaling followed by a translation. If $c\ne 0$, the polynomial-division identity

$$\varphi (z)=\frac{az+b}{cz+d}=\frac{a}{c}+\frac{bc-ad}{c}\cdot \frac{1}{cz+d}$$

decomposes $\varphi$ as: translate by $d$, scale by $c$, invert, scale by $(bc-ad)/c$, translate by $a/c$.

Conformality. The derivative is ${\varphi }^{\prime }(z)=(ad-bc)/(cz+d{)}^2$, nonzero on $\mathbb{C}\setminus \{-d/c\}$ (the chart at $\infty$ shows ${\varphi }^{\prime }$ also nonzero at $\infty$ and at $-d/c$ when computed in the appropriate chart). Hence $\varphi$ is conformal — angle-preserving — at every point of ${\mathbb{C}}_{\infty }$.

Counterexamples to common slips.

• Polynomials of degree $\ge 2$ are not Möbius transformations: $z\mapsto z^2$ is not bijective on ${\mathbb{C}}_{\infty }$ (it has degree $2$ as a branched cover of the sphere).
• The "transformation" with $ad-bc=0$ is not a Möbius transformation: the formula $(az+b)/(cz+d)$ is then constant (or undefined), and the bijectivity that defines the class fails.
• Anti-holomorphic maps such as $z\mapsto \bar{z}$ or $z\mapsto 1/\bar{z}$ are not Möbius transformations: they preserve circles-and-lines and unsigned angles, but they reverse orientation and are anti-holomorphic, not holomorphic.

Key theorem with proof [Intermediate+]

Theorem (three-point uniqueness). Given three distinct points $z_1,z_2,z_3\in {\mathbb{C}}_{\infty }$ and three distinct points $w_1,w_2,w_3\in {\mathbb{C}}_{\infty }$, there exists a unique Möbius transformation $\varphi :{\mathbb{C}}_{\infty }\to {\mathbb{C}}_{\infty }$ with $\varphi (z_i)=w_i$ for $i=1,2,3$. ^{[Ahlfors Ch. 3 §3]}

Proof. Existence. Define the cross-ratio of an ordered four-tuple of distinct points by

$$(z,z_1;z_2,z_3):=\frac{(z-z_2)(z_1-z_3)}{(z-z_3)(z_1-z_2)},$$

interpreted with the conventions $\infty /\infty =1$ and $0/0=1$ when one of $z,z_1,z_2,z_3$ is $\infty$ (so the corresponding factor is replaced by $1$). For fixed $z_1,z_2,z_3$ the assignment $T_{z_1{z}_2{z}_3}(z):=(z,z_1;z_2,z_3)$ is a Möbius transformation, since the numerator and denominator are linear in $z$ and the determinant $(z_1-z_3)(z_1-z_2)\cdot (-1)-(z_1-z_3)\cdot 1\cdot (-(z_1-z_2))\ne 0$ collapses to $-(z_1-z_2)(z_1-z_3)\ne 0$. Direct evaluation gives

$$T_{z_1{z}_2{z}_3}(z_1)=1,T_{z_1{z}_2{z}_3}(z_2)=0,T_{z_1{z}_2{z}_3}(z_3)=\infty .$$

Form the analogous Möbius transformation $T_{w_1{w}_2{w}_3}$. Then

$$\varphi :=T_{w_1{w}_2{w}_3}^{-1}\circ T_{z_1{z}_2{z}_3}$$

sends $z_i$ to $w_i$ for $i=1,2,3$. Existence holds.

Uniqueness. Suppose two Möbius transformations $\varphi$ and $\psi$ satisfy $\varphi (z_i)=w_i=\psi (z_i)$ for $i=1,2,3$. Then ${\psi }^{-1}\circ \varphi$ is a Möbius transformation fixing $z_1,z_2,z_3$. It suffices to show that any Möbius transformation $\rho$ fixing three distinct points is the identity.

Write $\rho (z)=(az+b)/(cz+d)$ with $ad-bc\ne 0$. The fixed-point equation $\rho (z)=z$ rearranges to $c{z}^2+(d-a)z-b=0$. This is a polynomial equation in $z$ of degree at most $2$. If $\rho$ has three distinct fixed points, the polynomial vanishes for three distinct values of $z$. A polynomial of degree at most $2$ with three distinct roots is identically zero, forcing $c=0$, $d=a$, and $b=0$. Thus $\rho (z)=az/a=z$, the identity.

The argument needs a small adjustment if one of the three fixed points is $\infty$: the condition $\rho (\infty )=\infty$ forces $c=0$, after which $\rho (z)=(a/d)z+(b/d)$ has the two remaining fixed points satisfying $(a/d-1)z+b/d=0$. Two distinct fixed points of a degree-$1$ polynomial in $z$ force the linear coefficient and the constant term to vanish: $a=d$ and $b=0$. So $\rho$ is the identity.

In either case ${\psi }^{-1}\circ \varphi =\mathrm{id}$, hence $\varphi =\psi$. $\square$

The proof simultaneously establishes the existence of the cross-ratio Möbius transformation and the uniqueness of the resulting map.

Bridge. Three-point uniqueness is the structural pivot from which the rest of the theory unfolds. First, it makes the cross-ratio a complete Möbius invariant of an ordered $4$-tuple: the unique Möbius taking three points to $1,0,\infty$ exists, so the value at the fourth point is well-defined and depends only on the four-tuple, and any Möbius invariant of a four-tuple is a function of the cross-ratio. Second, it lets the Möbius group act triply transitively on ${\mathbb{C}}_{\infty }$, with the stabiliser of three points reduced to the identity — exactly the regularity needed to identify ${\mathrm{PSL}}_2(\mathbb{C})$ with the full automorphism group of the sphere 06.01.07. Third, it builds toward the Riemann mapping theorem 06.01.06, where Möbius transformations supply the unique disk automorphism witnessing the uniqueness clause. Fourth, it shows that ${\mathrm{PSL}}_2(\mathbb{R})$ acts simply transitively on triples of boundary points of the upper half-plane, which is the structural reason hyperbolic geometry inherits the cross-ratio as its angle-and-distance bookkeeping.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none. Mathlib provides Matrix.GeneralLinearGroup, Matrix.SpecialLinearGroup, the abstract projectivisation Projectivization, and (with some assembly) the projective special linear group as a quotient, but the explicit fractional-linear action $\varphi :{\mathrm{GL}}_2(\mathbb{C})\to \mathrm{Aut}({\mathbb{C}}_{\infty })$ on the Riemann sphere, the kernel-of-scalars identification with ${\mathrm{PSL}}_2(\mathbb{C})$, the cross-ratio as a projective invariant, the trace-squared classification (parabolic / elliptic / hyperbolic / loxodromic), and the action of ${\mathrm{PSL}}_2(\mathbb{R})$ on the upper half-plane as orientation-preserving hyperbolic isometries are not formalised. See the lean_mathlib_gap field in this unit's frontmatter for the precise contribution roadmap.

Advanced results [Master]

The automorphism group. $\mathrm{Aut}({\mathbb{C}}_{\infty })={\mathrm{PSL}}_2(\mathbb{C})$. Every biholomorphism of the [Riemann sphere]06.01.07 to itself is a Möbius transformation: a holomorphic self-map of ${\mathbb{C}}_{\infty }$ is rational, bijectivity forces degree $1$, and a degree-$1$ rational function is exactly a Möbius transformation. The action is triply transitive on ${\mathbb{C}}_{\infty }$, and the stabiliser of any three distinct points reduces to the identity.

Cross-ratio and projective invariants. The cross-ratio $(z_1,z_2;z_3,z_4):=(z_1-z_3)(z_2-z_4)/((z_1-z_4)(z_2-z_3))$ is the unique Möbius invariant of an ordered $4$-tuple of distinct points, in the sense that two ordered $4$-tuples are Möbius-equivalent iff their cross-ratios coincide. Permuting the four arguments produces six values $\{\lambda ,1-\lambda ,1/\lambda ,1/(1-\lambda ),\lambda /(\lambda -1),(\lambda -1)/\lambda \}$ in general position, forming an action of the symmetric group $S_3$ on the cross-ratio (the Klein four-group acts as the identity). The locus where the cross-ratio is real corresponds to four points on a common circle-and-line — a synthetic-geometry statement equivalent to Ptolemy's theorem in the plane. ^{[Ahlfors Ch. 3 §3]}

Classification by trace squared. Let $\varphi \in {\mathrm{PSL}}_2(\mathbb{C})$ have a determinant-$1$ matrix representative $M$, and define ${\mathrm{tr}}^2(\varphi ):=\mathrm{tr}(M{)}^2$ — well-defined modulo the sign-flip $M\mapsto -M$. Provided $\varphi$ is not the identity, $\varphi$ is conjugate (in ${\mathrm{PSL}}_2(\mathbb{C})$) to one of four normal forms:

• Elliptic, ${\mathrm{tr}}^2\in [0,4)$: $\varphi$ has two distinct fixed points and is conjugate to $z\mapsto e^{i\theta }z$ with $\theta \in (0,2\pi )\setminus \{\pi$ paired with $-\pi \}$. Acts as a rotation about the two fixed points; orbits are circles.
• Parabolic, ${\mathrm{tr}}^2=4$: $\varphi$ has a single fixed point and is conjugate to the translation $z\mapsto z+1$. Acts as a "boundary translation"; orbits accumulate at the single fixed point.
• Hyperbolic, ${\mathrm{tr}}^2\in (4,\infty )$: $\varphi$ has two distinct fixed points and is conjugate to the real dilation $z\mapsto \lambda z$ with $\lambda >1$ real. Acts as a stretch from one fixed point (repelling) to the other (attracting); orbits are arcs of circles through both fixed points.
• Loxodromic, ${\mathrm{tr}}^2\in \mathbb{C}\setminus \mathbb{R}$: $\varphi$ has two distinct fixed points and is conjugate to $z\mapsto \lambda z$ for $\lambda \in \mathbb{C}$ with $|\lambda |\ne 1$ and $\lambda \notin \mathbb{R}$. Acts as a helical motion combining hyperbolic stretch with elliptic rotation; orbits are loxodromes (rhumb-line spirals) on the sphere.

Hyperbolic transformations are sometimes treated as a sub-case of loxodromic; the ${\mathrm{tr}}^2$ axis cleanly separates the two when one insists on the real-positive-$\lambda$ normal form.

Circle-and-line preservation. Möbius transformations send the family of generalised circles in ${\mathbb{C}}_{\infty }$ — the union of Euclidean circles in $\mathbb{C}$ and Euclidean straight lines (regarded as circles through $\infty$) — to itself, and act transitively on this family. In the stereographic identification with $S^2$, generalised circles correspond exactly to circles on the sphere (intersections with planes), and the Möbius group acts as the conformal group of $S^2$.

Hyperbolic geometry. The subgroup ${\mathrm{PSL}}_2(\mathbb{R})\subset {\mathrm{PSL}}_2(\mathbb{C})$ — Möbius transformations with real coefficients — preserves the upper half-plane $\mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$ and acts on it as the group of orientation-preserving isometries of the Poincaré metric $d{s}^2=(d{x}^2+d{y}^2)/y^2$. The geodesics of this metric are vertical lines and Euclidean semicircles meeting the real axis perpendicularly — precisely the generalised circles in ${\mathbb{C}}_{\infty }$ that meet the real axis at right angles. The disk model $\mathbb{D}$ is Möbius-equivalent to $\mathbb{H}$ via the Cayley transform $z\mapsto (z-i)/(z+i)$, and the disk's hyperbolic isometry group is $\mathrm{PSU}(1,1)\cong {\mathrm{PSL}}_2(\mathbb{R})$.

Modular group. The further subgroup ${\mathrm{PSL}}_2(\mathbb{Z})\subset {\mathrm{PSL}}_2(\mathbb{R})$ — Möbius transformations with integer coefficients — is the modular group. It acts on $\mathbb{H}$ as a discrete group of hyperbolic isometries, with fundamental domain the standard region $\{|z|\ge 1,|\mathrm{Re}z|\le 1/2\}$. The quotient $\mathbb{H}/{\mathrm{PSL}}_2(\mathbb{Z})$ parametrises elliptic curves over $\mathbb{C}$ up to isomorphism, with the $j$-invariant supplying the explicit holomorphic identification with ${\mathbb{C}}_{\infty }$ minus a cusp. The modular group is the gateway to the theory of modular forms and to the moduli of complex tori.

Synthesis. The Möbius group is the foundational symmetry group of one-dimensional complex analysis, and the synthesis runs in four directions. First, the synthesis with [Riemann-sphere geometry]06.01.07: ${\mathrm{PSL}}_2(\mathbb{C})$ is precisely $\mathrm{Aut}({\mathbb{C}}_{\infty })$, the conformal group of the round sphere under stereographic identification, and the chordal metric on ${\mathbb{C}}_{\infty }$ is the unique Möbius-quasi-invariant Riemannian metric up to scale. Second, the synthesis with the [Riemann mapping theorem]06.01.06: every simply-connected proper subset of $\mathbb{C}$ is biholomorphic to $\mathbb{D}$, and the biholomorphism is unique up to a Möbius automorphism of $\mathbb{D}$ — Möbius transformations supply the automorphism group that the Riemann mapping theorem's uniqueness clause measures by. Third, the synthesis with hyperbolic geometry: ${\mathrm{PSL}}_2(\mathbb{R})$ on $\mathbb{H}$ is the orientation-preserving isometry group of the Poincaré half-plane, and every hyperbolic surface arises as $\mathbb{H}/\Gamma$ for $\Gamma \subset {\mathrm{PSL}}_2(\mathbb{R})$ a Fuchsian group. Fourth, the synthesis with arithmetic and modular forms: ${\mathrm{PSL}}_2(\mathbb{Z})$ on $\mathbb{H}$ presents the moduli space of elliptic curves and feeds the entire theory of modular forms, the $j$-invariant, the Eisenstein series, and the trace formulas of Selberg. The Möbius group connects classical complex analysis to algebraic geometry (via ${\mathbb{P}}^1$ and the cross-ratio), to differential geometry (via hyperbolic isometry), and to number theory (via the modular group) in a single linear-fractional package.

Full proof set [Master]

The advanced results assembled above follow from the proofs given in earlier sections together with classical complex-analytic and group-theoretic arguments, recorded in Ahlfors Complex Analysis Ch. 3 §3 and Beardon The Geometry of Discrete Groups Chs. 3–4. Three reference points: (a) the identification $\mathrm{Aut}({\mathbb{C}}_{\infty })={\mathrm{PSL}}_2(\mathbb{C})$ uses the rationality of holomorphic self-maps of ${\mathbb{C}}_{\infty }$ — already established for the [Riemann sphere]06.01.07 — together with the bijectivity-forces-degree-$1$ argument: a degree-$d$ rational map ${\mathbb{C}}_{\infty }\to {\mathbb{C}}_{\infty }$ is $d$-to-$1$ generically, so $d=1$ for a bijection. (b) The trace-squared classification follows by reducing $\varphi$ to Jordan normal form: a determinant-$1$ matrix in ${\mathrm{SL}}_2(\mathbb{C})$ has eigenvalues $\lambda ,{\lambda }^{-1}$ with ${\mathrm{tr}}^2=(\lambda +{\lambda }^{-1}{)}^2$; the four cases for ${\mathrm{tr}}^2$ correspond to $\lambda$ on the unit circle (elliptic), $\lambda =\pm 1$ (parabolic, the non-diagonalisable Jordan block), $\lambda >0$ real (hyperbolic), or $\lambda$ generic complex (loxodromic). (c) The hyperbolic-isometry interpretation of ${\mathrm{PSL}}_2(\mathbb{R})$ on $\mathbb{H}$ follows by direct computation of the pullback of $d{s}^2=(d{x}^2+d{y}^2)/y^2$ under $z\mapsto (az+b)/(cz+d)$ with $a,b,c,d\in \mathbb{R},ad-bc=1$: the Jacobian computation shows the metric is invariant. The closing identification of $\mathbb{H}/{\mathrm{PSL}}_2(\mathbb{Z})$ with the moduli of elliptic curves uses the lattice description of complex tori and is detailed in dedicated jacobians and modular-form units. ^{[Ahlfors Ch. 3 §3; Beardon Chs. 3–4]}

Connections [Master]

• Riemann sphere 06.01.07 — The Riemann sphere supplies the underlying space on which the Möbius group acts, and the identification $\mathrm{Aut}({\mathbb{C}}_{\infty })={\mathrm{PSL}}_2(\mathbb{C})$ links the unit's group-theoretic side to the sphere's complex-manifold side. The chart $z\mapsto 1/z$ used to make ${\mathbb{C}}_{\infty }$ a Riemann surface is itself the Möbius inversion that ties the two charts together; the holomorphic atlas and the linear-fractional group are two faces of one structure.

• Riemann mapping theorem 06.01.06 — Every simply-connected proper subset of $\mathbb{C}$ is biholomorphic to the unit disk, and the biholomorphism is unique up to a Möbius automorphism of $\mathbb{D}$. Möbius transformations supply the precise group measuring the uniqueness clause, and the explicit form of disk automorphisms (Exercise 5) is the standard normalisation used to fix a basepoint and a tangent direction.

• Holomorphic function 06.01.01 — Conformality and holomorphicity are equivalent at each point of nonzero derivative, and Möbius transformations are the simplest non-constant holomorphic maps preserving the family of generalised circles. They serve as the local model for biholomorphism between domains and as the test cases for every conformal-mapping theorem.

• Cauchy integral formula 06.01.02 — Möbius transformations preserve the contour-integration framework: the change of variables $w=\varphi (z)$ transports an integration cycle through a Möbius-equivalent domain, and the residue at the pole $-d/c$ of $\varphi$ produces the correct boundary contribution under inversion. The cross-ratio appears naturally in evaluating contour integrals over arcs of generalised circles.

• Meromorphic function 06.01.05 — Möbius transformations are the meromorphic functions on ${\mathbb{C}}_{\infty }$ of degree $1$ — exactly the rational functions $P(z)/Q(z)$ with $\mathrm{deg}P=\mathrm{deg}Q=1$ and no common factor. The classification of meromorphic self-maps of the sphere by degree begins with the Möbius case and ascends through Blaschke products to the full theory of rational dynamics.

Historical & philosophical context [Master]

August Ferdinand Möbius introduced the linear-fractional transformations and their inversive geometry in Die Theorie der Kreisverwandtschaft in rein geometrischer Darstellung (1855), building on his 1827 work on barycentric calculus and the projective treatment of geometry. Möbius's Kreisverwandtschaft — "circle relationship" or, more loosely, "the kinship of circles" — was the synthetic-geometry observation that inversion in a circle, together with the rigid motions, generates a group preserving the family of circles-and-lines; the algebraic form $z\mapsto (az+b)/(cz+d)$ is the analytic transcription of that synthetic group on ${\mathbb{C}}_{\infty }$. ^{[Möbius 1855]}

Felix Klein's Erlanger Programm (1872) reorganised geometry around group actions, with Möbius geometry — the theory of ${\mathrm{PGL}}_2(\mathbb{C})$ acting on ${\mathbb{C}}_{\infty }$ — taking pride of place as the canonical conformal Klein geometry. Klein's framework absorbed Möbius's Kreisverwandtschaft, Cayley's projective metric, and the emerging hyperbolic-geometry programme into a single category of "geometries as quotients $G/H$". ^{[Klein 1872 *Erlanger Programm*]}

Henri Poincaré in his Mémoire sur les groupes fuchsiens (Acta Mathematica 1, 1882) used ${\mathrm{PSL}}_2(\mathbb{R})$ acting on $\mathbb{H}$ to construct the modern theory of automorphic forms and hyperbolic manifolds. Poincaré realised that the discrete subgroups of ${\mathrm{PSL}}_2(\mathbb{R})$ — the Fuchsian groups — supply the universal-cover structure for hyperbolic Riemann surfaces, completing the trichotomy with the elliptic (${\mathbb{C}}_{\infty }/\{1\}$) and parabolic ($\mathbb{C}/\Lambda$) cases. The standard pedagogical exposition was codified by Lars Ahlfors in Complex Analysis (1953; 3rd ed. 1979) Ch. 3, with the matrix-presentation of the group and the trace-squared classification given the formal treatment now standard.

Bibliography [Master]

• Möbius, A. F., Die Theorie der Kreisverwandtschaft in rein geometrischer Darstellung, Abh. Königl. Sächs. Ges. Wiss., Math.-Phys. Kl. 2 (1855), 529–595.
• Klein, F., Vergleichende Betrachtungen über neuere geometrische Forschungen (Erlanger Programm), Erlangen (1872); reprinted Math. Ann. 43 (1893), 63–100.
• Poincaré, H., Théorie des groupes fuchsiens, Acta Math. 1 (1882), 1–62.
• Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw-Hill (1979). Ch. 3 §3.
• Beardon, A. F., The Geometry of Discrete Groups, GTM 91, Springer (1983). Chs. 3–4.
• Ford, L. R., Automorphic Functions, McGraw-Hill (1929); reprinted AMS Chelsea (1972).
• Needham, T., Visual Complex Analysis, Oxford University Press (1997). Ch. 3.
• Schwerdtfeger, H., Geometry of Complex Numbers: Circle Geometry, Möbius Transformation, Non-Euclidean Geometry, Dover (1979).