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Hyperbolic Riemann Surfaces and Uniformization: Examples and Counterexamples

1 · Prerequisites

2 · Summary

These examples compute the hyperbolic metric and its geodesics in the two standard models, exhibit the three universal-covering types, and follow the type through explicit covering constructions. The Cayley map carries the disc metric $2 ∣ d z ∣ / (1 - ∣ z ∣^{2})$ to $∣ d w ∣ / Im w$ , the radial disc segment and the vertical half-plane ray realise the distance values $2 artanh r$ and $∣ \log y ∣$ , and the geodesics are the Euclidean circles and lines meeting the boundary orthogonally.

Compactness and Liouville separate the models: the sphere is compact while the plane and the disc are not, and a biholomorphism of the plane onto the disc would be a bounded nonconstant entire function, which Liouville's theorem excludes. The exponential map makes the annulus and the punctured disc quotients of simply connected half-plane and strip domains with deck group generated by translation by $2 π i$ , and a rank-two lattice of translations gives the complex torus with parabolic universal-covering type and genus one.

The final example completes the hyperelliptic curve $y^{2} = \prod_{j = 1}^{6} (x - a_{j})$ at infinity with two unramified points, computes its genus two by Riemann-Hurwitz, and transfers its disc uniformization to a cocompact Fuchsian quotient: the deck group is a torsion-free discrete subgroup of the disc automorphisms acting freely and properly discontinuously with compact quotient. Each example states its own choice assumptions and uses only the cited suppliers; no Gauss-Bonnet theorem is appealed to for the genus computation.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Hyperbolic distances and geodesics in disc and half-plane

Example

Write $D = {∣ z ∣ < 1}$ and $H = {Im w > 0}$ , and define the Cayley map and its inverse by

$C (w) : = \frac{w - i}{w + i} (w \in C ∖ {- i}), C^{- 1} (ζ) : = i \frac{1 + ζ}{1 - ζ} (ζ \in C ∖ {1}) .$

Then the following four statements hold.

$C$ maps $H$ biholomorphically onto $D$ . Pushing the Poincare metric $2 ∣ d z ∣ / (1 - ∣ z ∣^{2})$ of $D$ forward along $C^{- 1}$ turns it into the metric $∣ d w ∣ / Im w$ on $H$ : writing $ℓ_{H}$ for the lengths computed with that metric and $d_{H}$ for the associated infimum distance, one has $d_{H} (p, q) = d_{D} (C (p), C (q))$ .
For $0 \leq r < 1$ the radial segment from $0$ to $r$ attains $d_{D} (0, r) = 2 artanh r$ , and for $y > 0$ the vertical segment from $i$ to $i y$ attains $d_{H} (i, i y) = ∣ \log y ∣$ .
The geodesic segment joining distinct $z, w \in D$ is the subarc with endpoints $z, w$ inside $D$ of a Euclidean circle or line that meets the unit circle at right angles; concretely it is the image of the radial segment from $0$ to $φ_{z} (w)$ under the disc automorphism $φ_{z}$ .
Likewise the geodesic segment joining distinct $p, q \in H$ is the subarc with endpoints $p, q$ inside $H$ of a vertical line or of a Euclidean circle with centre on the real axis.

Facts & Assumptions

Given: The unit disc $D$ , the upper half-plane $H$ , the Cayley map $C$ with its inverse, and the Poincare metric of $D$ .

[F1]

On $D$ the Poincare metric is $2 ∣ d z ∣ / (1 - ∣ z ∣^{2})$ ; the Poincare length of a piecewise $C^{1}$ curve $γ : [a, b] \to D$ is $ℓ_{D} (γ) = \int_{a}^{b} 2 ∣ γ^{'} (t) ∣ / (1 - ∣ γ (t) ∣^{2}) d t$ , and $d_{D} (z, w)$ is the infimum of these lengths over piecewise $C^{1}$ curves from $z$ to $w$ (The Poincare metric and distance on the unit disc).

[F2]

For $z, w \in D$ one has $d_{D} (z, w) = 2 artanh ∣ φ_{z} (w) ∣$ , and every automorphism of $D$ preserves $d_{D}$ (The Poincare distance has the formula $2 artanh ∣ φ_{z} (w) ∣$ and is disc-automorphism invariant).

[F3]

The disc is $D = {z : ∣ z ∣ < 1}$ , the upper half-plane is $H = {z : Im z > 0}$ , and the Blaschke factor is $φ_{a} (z) = (a - z) / (1 - \overline{a} z)$ with denominator nonzero on $D$ ; moreover $φ_{a} (0) = a$ and $φ_{a} (a) = 0$ (The unit disc, the upper half-plane, and Blaschke factors).

[F4]

For every $a \in D$ the Blaschke factor satisfies $φ_{a} (D) = D$ and $φ_{a} (φ_{a} (z)) = z$ for $z \in D$ , and $φ_{a}$ is an automorphism of $D$ (Blaschke factors are automorphisms of the disc).

[F5]

A map $f : H \to H$ is an automorphism of $H$ if and only if $f (z) = (a z + b) / (c z + d)$ with $a, b, c, d \in R$ and $a d - b c > 0$ (Automorphisms of the upper half-plane are real Mobius maps).

[F6]

For $∣ u ∣ < 1$ one has $artanh u = \frac{1}{2} \log \frac{1 + u}{1 - u}$ (Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains).

[F7]

The sum, product and quotient rules hold for complex derivatives, and the reciprocal and quotient formulas are $(1 / g)^{'} (a) = - g^{'} (a) / g (a)^{2}$ and $(f / g)^{'} (a) = (f^{'} (a) g (a) - f (a) g^{'} (a)) / g (a)^{2}$ wherever $g (a) \neq 0$ (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[F8]

If $f : U \to V$ and $g : V \to C$ are complex differentiable at $a$ and $f (a)$ respectively, then $(g \circ f)^{'} (a) = g^{'} (f (a)) f^{'} (a)$ (The chain rule for complex derivatives).

[F9]

For $x > 0$ the real logarithm is differentiable with $\log^{'} (x) = 1 / x$ , and $\log x = \int_{1}^{x} d t / t$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

Proof technique: direct computation: differentiate the Cayley map, transport lengths, and identify the minimizers through two one-dimensional monotonicity inequalities with their equality cases.

Verification

1.1F3givenalgebra

For every $w \in C$ one has the identity $∣ w + i ∣^{2} - ∣ w - i ∣^{2} = 4 Im w$ ; hence for $w \in H$ , $∣ C (w) ∣^{2} = ∣ w - i ∣^{2} / ∣ w + i ∣^{2} < 1$ , and for $∣ ζ ∣ < 1$ the point $w : = C^{- 1} (ζ)$ satisfies $Im w = (1 - ∣ ζ ∣^{2}) / ∣ 1 - ζ ∣^{2} > 0$ . Direct substitution gives $C (C^{- 1} (ζ)) = ζ$ and $C^{- 1} (C (w)) = w$ , so $C$ is a bijection of $H$ onto $D$ .

1.2F1F2F3F6F7algebra

The radial segment $γ (t) = t$ , $0 \leq t \leq r$ , from $0$ to $r$ has, by [F6] and [F7], $ℓ_{D} (γ) = \int_{0}^{r} \frac{2 d t}{1 - t^{2}} = 2 artanh r$ , since $\frac{d}{d t} 2 artanh t = \frac{2}{1 - t^{2}}$ and $2 artanh 0 = 0$ . On the other hand [F3] gives $φ_{0} (r) = (0 - r) / (1 - 0) = - r$ , so [F2] gives $d_{D} (0, r) = 2 artanh ∣ - r ∣ = 2 artanh r$ . Hence the radial segment attains the distance.

1.3F1F2F6F8givenalgebra

Let $u \in D ∖ {0}$ and let $γ : [a, b] \to D$ be piecewise $C^{1}$ with $γ (a) = 0$ and $γ (b) = u$ ; put $ρ : = ∣ γ ∣$ and $f : = 2 artanh ρ$ . At every point where both derivatives exist one has $∣ ρ^{'} ∣ \leq ∣ γ^{'} ∣$ , and $f$ is absolutely continuous with $∣ f^{'} ∣ = 2 ∣ ρ^{'} ∣ / (1 - ρ^{2})$ , so $ℓ_{D} (γ) = \int_{a}^{b} 2 ∣ γ^{'} ∣ / (1 - ρ^{2}) d t \geq \int_{a}^{b} 2 ∣ ρ^{'} ∣ / (1 - ρ^{2}) d t \geq ∣ \int_{a}^{b} f^{'} ∣ = ∣ f (b) - f (a) ∣ = 2 artanh ∣ u ∣$ . By [F2] the last quantity is $d_{D} (0, u)$ , so every curve from $0$ to $u$ has length at least $2 artanh ∣ u ∣$ .

1.4F3F4algebra

The Blaschke factor satisfies $φ_{a}^{'} (v) = - (1 - ∣ a ∣^{2}) / (1 - \overline{a} v)^{2}$ and $1 - ∣ φ_{a} (v) ∣^{2} = (1 - ∣ a ∣^{2}) (1 - ∣ v ∣^{2}) / ∣ 1 - \overline{a} v ∣^{2}$ for $v \in D$ ; hence $2 ∣ φ_{a}^{'} (v) ∣ / (1 - ∣ φ_{a} (v) ∣^{2}) = 2 / (1 - ∣ v ∣^{2})$ , so $ℓ_{D} (φ_{a} \circ γ) = ℓ_{D} (γ)$ for every piecewise $C^{1}$ curve $γ$ in $D$ .

1.5F3F4algebra

Fix $a \in D$ . If $a = 0$ , put $S_{0} : = R \cup {\infty}$ ; since $φ_{0} (t) = - t$ , this is the extended image of the real line. If $a \neq 0$ , put $S_{a} : = {φ_{a} (t) : t \in R, 1 - \overline{a} t \neq 0} \cup {1 / \overline{a}}$ . Then $S_{a}$ is a Euclidean circle or line meeting the unit circle at right angles. Indeed, since $φ_{a}$ is an involution [F4], for $u \neq 1 / \overline{a}$ one has $u \in S_{a}$ if and only if $φ_{a} (u) \in R$ , and expanding $φ_{a} (u) = (a - u) / (1 - \overline{a} u)$ gives $2 i Im φ_{a} (u) = [(a - u) (1 - a \overline{u}) - (\overline{a} - \overline{u}) (1 - \overline{a} u)] / ∣ 1 - \overline{a} u ∣^{2} = [(a - \overline{a}) (1 + ∣ u ∣^{2}) + (1 - a^{2}) \overline{u} - (1 - {\overline{a}}^{2}) u] / ∣ 1 - \overline{a} u ∣^{2}$ , so $u \in S_{a}$ if and only if $(a - \overline{a}) (1 + ∣ u ∣^{2}) + (1 - a^{2}) \overline{u} - (1 - {\overline{a}}^{2}) u = 0$ . If $a \in R$ , this reads $(1 - a^{2}) (\overline{u} - u) = 0$ , so $S_{a} = \hat{R}$ is the real line, which meets the unit circle at $\pm 1$ at right angles. If $a \notin R$ , dividing by $a - \overline{a} = 2 i Im a$ and writing $A : = (1 - a^{2}) / (a - \overline{a})$ turns the equation into $1 + ∣ u ∣^{2} + A \overline{u} + \overline{A} u = 0$ , that is $∣ u + A ∣^{2} = ∣ A ∣^{2} - 1$ . Since $∣ A ∣^{2} - 1 = (∣ 1 - a^{2} ∣^{2} - ∣ a - \overline{a} ∣^{2}) / ∣ a - \overline{a} ∣^{2} = (1 - ∣ a ∣^{2})^{2} / ∣ a - \overline{a} ∣^{2} > 0$ , this is a circle with centre $- A$ and radius $ρ > 0$ ; a point $u \neq 1 / \overline{a}$ lies on that circle exactly when $u \in S_{a}$ , and $1 / \overline{a} \in S_{a}$ is the limit $\lim_{∣ t ∣ \to \infty} φ_{a} (t)$ of points of the circle, so $S_{a}$ is exactly this circle. Finally $∣ - A ∣^{2} - ρ^{2} = 1$ , and a circle with centre $c$ and radius $ρ$ that meets the unit circle meets it orthogonally precisely when $∣ c ∣^{2} - ρ^{2} = 1$ : at an intersection point $w$ the tangents are perpendicular exactly when the radius vectors $w$ and $w - c$ are perpendicular, that is $Re (\overline{w} (w - c)) = 0$ , which with $∣ w ∣ = 1$ says $Re (\overline{w} c) = 1$ , and substituting this into $∣ w - c ∣^{2} = ρ^{2}$ , namely $1 - 2 Re (\overline{w} c) + ∣ c ∣^{2} = ρ^{2}$ , gives $∣ c ∣^{2} = ρ^{2} + 1$ . So $S_{a}$ meets the unit circle at right angles.

1.6F5algebra

Let $M (z) = (a z + b) / (c z + d)$ have real $a, b, c, d$ with $a d - b c > 0$ . For $s \in R$ one computes $M (i s) = (b + a i s) / (d + c i s) = (b d + a c s^{2} + i s (a d - b c)) / (c^{2} s^{2} + d^{2})$ . If $c = 0$ , then $Re M (i s) = b / d$ is constant and the image of the imaginary line is the vertical line $Re w = b / d$ . If $c \neq 0$ and $d = 0$ , then $Re M (i s) = a / c$ is constant and the image is the vertical line $Re w = a / c$ . If $c \neq 0$ and $d \neq 0$ , then multiplying out shows that every $M (i s)$ satisfies $(X - p)^{2} + Y^{2} = σ^{2}$ with $X = Re M (i s)$ , $Y = Im M (i s)$ , centre $p : = \frac{1}{2} (\frac{a}{c} + \frac{b}{d}) \in R$ and radius $σ = ∣ a d - b c ∣ / (2 ∣ c ∣ ∣ d ∣) > 0$ ; the real points $a / c$ and $b / d$ of this circle are $p \pm σ$ , so it is a Euclidean circle with centre on $R$ . In every case the image of the imaginary line meets $R$ at right angles: vertical lines do so plainly, and a circle with centre on $R$ has vertical tangents at its two real points.

2.1F7step 1.1algebra

For $w \in H$ the quotient rule [F7] gives $C^{'} (w) = 2 i / (w + i)^{2} \neq 0$ , and step 1.1 gives $1 - ∣ C (w) ∣^{2} = (∣ w + i ∣^{2} - ∣ w - i ∣^{2}) / ∣ w + i ∣^{2} = 4 Im w / ∣ w + i ∣^{2}$ ; therefore $2 ∣ C^{'} (w) ∣ / (1 - ∣ C (w) ∣^{2}) = (4 / ∣ w + i ∣^{2}) \cdot (∣ w + i ∣^{2} / (4 Im w)) = 1 / Im w$ .

2.2F1F4step 1.3algebra

In step 1.3 equality holds if and only if $γ$ is a monotone reparametrisation of the radial segment from $0$ to $u$ . Indeed, equality in the second inequality forces $f$ , hence $ρ$ , to be nondecreasing; equality in the first forces $∣ γ^{'} ∣ = ∣ ρ^{'} ∣$ almost everywhere, which combined with $ρ^{'} \geq 0$ gives $γ^{'} = λ γ$ with $λ \geq 0$ real wherever $γ \neq 0$ , so the unit vector $γ / ρ$ is constant on each interval on which $ρ > 0$ . Since $ρ$ is nondecreasing from $ρ (a) = 0$ to $ρ (b) = ∣ u ∣ > 0$ , the set ${ρ > 0}$ is an interval $(t_{0}, b]$ , and continuity of $γ / ρ$ there gives $γ (t) / ρ (t) = u / ∣ u ∣$ for $t > t_{0}$ . Thus $γ$ traces the segment $[0, u]$ with nondecreasing modulus, and conversely every such parametrisation realises equality.

2.3F3step 1.5algebra

Let $z \in D$ and let $e \in C$ with $∣ e ∣ = 1$ . For all $v \in C$ with $1 - \overline{z} e v \neq 0$ one has, multiplying numerator and denominator by $\overline{e}$ , $φ_{z} (e v) = (z - e v) / (1 - \overline{z} e v) = e (z \overline{e} - v) / (1 - \overline{z \overline{e}} v) = e φ_{z \overline{e}} (v)$ . Hence the image of the line ${t e : t \in R} \cup {\infty}$ under $φ_{z}$ is $e S_{z \overline{e}}$ , a rotation of the circle or line of step 1.5. Multiplication by $e$ preserves Euclidean circles and lines, fixes the unit circle, and carries a circle of centre $c$ and radius $ρ$ to the circle of centre $e c$ and radius $ρ$ , so $e S_{z \overline{e}}$ is again a Euclidean circle or line meeting the unit circle at right angles.

3.1F1F8step 1.1step 2.1

Define $ℓ_{H} (γ) : = \int_{a}^{b} ∣ γ^{'} (t) ∣ / Im γ (t) d t$ for piecewise $C^{1}$ curves $γ : [a, b] \to H$ , and $d_{H} (p, q) : = \inf ℓ_{H} (γ)$ over such curves from $p$ to $q$ . By the chain rule and step 2.1, $ℓ_{D} (C \circ γ) = ℓ_{H} (γ)$ for every piecewise $C^{1}$ curve $γ$ in $H$ , and likewise $ℓ_{H} (C^{- 1} \circ σ) = ℓ_{D} (σ)$ for every piecewise $C^{1}$ curve $σ$ in $D$ ; since $C$ and $C^{- 1}$ transport curves in both directions, taking infima gives $d_{H} (p, q) = d_{D} (C (p), C (q))$ for all $p, q \in H$ , and $C$ is a biholomorphism of $H$ onto $D$ .

3.2F2F4step 1.3step 1.4step 2.2algebra

Let $z, w \in D$ with $z \neq w$ , and put $φ : = φ_{z}$ and $u : = φ (w) \neq 0$ . By [F4], $φ$ is an automorphism of $D$ , $φ (z) = 0$ , and $φ^{- 1} = φ$ . A piecewise $C^{1}$ curve $σ$ from $z$ to $w$ has $ℓ_{D} (σ) = d_{D} (z, w)$ if and only if $φ \circ σ$ has length $d_{D} (0, u)$ , because step 1.4 gives $ℓ_{D} (φ \circ σ) = ℓ_{D} (σ)$ and [F2] gives $d_{D} (0, u) = d_{D} (φ (z), φ (w)) = d_{D} (z, w)$ . By steps 1.3 and 2.2 the length-minimising curves from $0$ to $u$ are exactly the monotone reparametrisations of the radial segment $[0, u]$ . Consequently the length-minimising curves from $z$ to $w$ are exactly the images under $φ$ of those curves, that is, the images of the radial segment from $0$ to $u = φ_{z} (w)$ .

4.1F3F6F9step 3.1step 1.2algebra

For $y > 0$ one has $C (i) = 0$ and $C (i y) = (i y - i) / (i y + i) = (y - 1) / (y + 1)$ , a real number of modulus $∣ y - 1 ∣ / (y + 1) < 1$ . Steps 3.1 and 1.2 and [F6] therefore give $d_{H} (i, i y) = d_{D} (0, ∣ y - 1 ∣ / (y + 1)) = 2 artanh (∣ y - 1 ∣ / (y + 1)) = ∣ \log y ∣$ : for $y \geq 1$ the logarithm formula gives $2 artanh \frac{y - 1}{y + 1} = \log \frac{(y + 1) + (y - 1)}{(y + 1) - (y - 1)} = \log y$ , and for $0 < y < 1$ replacing $y$ by $1 / y$ gives the same identity with $\log (1 / y) = ∣ \log y ∣$ . The vertical segment $γ (t) = i t$ , $t$ running from $1$ to $y$ , has $ℓ_{H} (γ) = ∣ \int_{1}^{y} d s / s ∣ = ∣ \log y ∣$ by [F9], so it attains the distance.

4.2F3step 3.2step 2.3algebra

Now let $z, w \in D$ with $z \neq w$ , put $u : = φ_{z} (w) \neq 0$ and $e : = u / ∣ u ∣$ . The radial segment $[0, u]$ is contained in the line ${t e : t \in R}$ , so by step 3.2 the length-minimising curves from $z$ to $w$ are the images under $φ_{z}$ of that segment, and these lie on $φ_{z} ({t e} \cup {\infty})$ , which by step 2.3 is a Euclidean circle or line meeting the unit circle at right angles.

5.1F9step 3.1step 4.1algebra

Let $y > 0$ and let $γ : [a, b] \to H$ be piecewise $C^{1}$ from $i$ to $i y$ ; put $ρ : = Im γ > 0$ and $g : = \log ρ$ . Then $∣ ρ^{'} ∣ \leq ∣ γ^{'} ∣$ wherever both derivatives exist, so $ℓ_{H} (γ) = \int_{a}^{b} ∣ γ^{'} ∣ / ρ d t \geq \int_{a}^{b} ∣ ρ^{'} ∣ / ρ d t \geq ∣ \int_{a}^{b} g^{'} ∣ = ∣ \log y - \log 1 ∣ = ∣ \log y ∣$ , with $g$ absolutely continuous by [F9]. Equality in the second inequality forces $ρ$ to be monotone, and equality in the first (where $∣ γ^{'} ∣^{2} = (Re γ^{'})^{2} + ρ^{' 2}$ ) forces $x : = Re γ$ to satisfy $x^{'} = 0$ almost everywhere, hence to be constant; so the curves attaining $∣ \log y ∣$ are exactly the monotone parametrisations of the vertical segment from $i$ to $i y$ . In particular the vertical segment is the unique minimiser, and step 4.1 shows its length is $d_{H} (i, i y) = ∣ \log y ∣$ .

6.1F3F4F5F7step 1.1step 3.1step 5.1algebra

Let $p, q \in H$ with $p \neq q$ , put $z^{'} : = C (p)$ , $w^{'} : = C (q)$ , $u^{'} : = φ_{z^{'}} (w^{'})$ and $r : = ∣ u^{'} ∣ \in (0, 1)$ . Let $R (ζ) : = ζ \overline{u^{'}} / r$ be the rotation of $D$ carrying $u^{'}$ to $r$ , and put $g : = R \circ φ_{z^{'}}$ and $m : = C^{- 1} \circ g \circ C$ . Then $g$ is an automorphism of $D$ with $g (z^{'}) = 0$ and $g (w^{'}) = r$ , so $m$ is a biholomorphic self-map of $H$ with $m (p) = C^{- 1} (0) = i$ and $m (q) = C^{- 1} (r) = i (1 + r) / (1 - r) = : i y$ with $y > 0$ . By [F5] one has $m (z) = (a z + b) / (c z + d)$ with $a, b, c, d$ real and $a d - b c > 0$ , and likewise $m^{- 1} (z) = (d z - b) / (- c z + a)$ has real coefficients and determinant $a d - b c > 0$ . For such a map the quotient rule [F7] gives $m^{'} (z) = (a d - b c) / (c z + d)^{2}$ and $Im m (z) = ((a d - b c) Im z) / ∣ c z + d ∣^{2}$ , hence $∣ m^{'} (z) ∣ / Im m (z) = 1 / Im z$ for $z \in H$ ; so $m$ preserves $ℓ_{H}$ and transports minimisers to minimisers.

7.1F5step 6.1step 1.6

By steps 5.1 and 6.1 the minimisers from $p$ to $q$ are the images under $m^{- 1}$ of the monotone parametrisations of the vertical segment from $i$ to $i y$ , and these images lie on $m^{- 1} ({i s : s \in R} \cup {\infty})$ . Since $m^{- 1}$ is again given by real Mobius coefficients with positive determinant, step 1.6 shows that this set is a vertical line or a Euclidean circle with centre on the real axis, meeting $R$ at right angles. So the geodesic segment from $p$ to $q$ is an arc of such a circle or line.

8.1step 3.1step 1.2step 4.1step 2.2step 4.2step 5.1step 7.1∎

Steps 3.1, 1.2, 4.1, 4.2 and 7.1 establish the four clauses of the Example. All curves and maps in the argument are given by explicit formulae; in particular the disc and half-plane geodesics are obtained by inverting explicit biholomorphisms, and the infima in [F1] and step 3.1 are taken over explicitly parametrised families, so no choice principle is used. When $z = w$ in $D$ or $p = q$ in $H$ the constant curve has length $0 = d_{D} (z, z) = d_{H} (p, p)$ , and steps 2.2 and 5.1 identify the strict minimisers only in the nondegenerate case.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Annulus and punctured disc have hyperbolic universal covers

Example

Assume the Axiom of Choice. Fix $0 < r < 1$ and put $L : = - \log r > 0$ , and consider the vertical strip, the finite annulus, the left half-plane and the punctured disc

$S_{r} : = {w \in C : \log r < Re w < 0}, A_{r} : = {z \in C : r < ∣ z ∣ < 1} = A (0; r, 1),$

$H_{-} : = {w \in C : Re w < 0}, D^{*} : = {z \in C : 0 < ∣ z ∣ < 1} = A (0; 0, 1) .$

Then the following hold.

$\exp : S_{r} \to A_{r}$ and $\exp : H_{-} \to D^{*}$ are covering maps, and $\exp^{- 1} (A_{r}) = S_{r}$ , $\exp^{- 1} (D^{*}) = H_{-}$ : the exponential maps the vertical strip onto the finite annulus and the left half-plane onto the punctured disc.
$Deck (\exp ∣_{S_{r}}) = {w \mapsto w + 2 π i k : k \in Z}$ and $Deck (\exp ∣_{H_{-}}) = {w \mapsto w + 2 π i k : k \in Z}$ : both deck groups are infinite cyclic, generated by the translation $τ (w) = w + 2 π i$ .
$S_{r}$ and $H_{-}$ are biholomorphic to $D$ ; consequently both are simply connected, the two exponential maps are universal covering spaces, and $A_{r}$ and $D^{*}$ have hyperbolic universal-covering type (Spherical, parabolic and hyperbolic universal-covering types).

On the modulus contrast. The finite annulus $A_{r}$ carries the modulus parameter $(- \log r) / (2 π)$ determined by its inner radius, while the punctured disc $D^{*}$ has a cusp end at the puncture; the two surfaces are homeomorphic, and this example does not attempt to prove that they are non-biholomorphic, which needs a conformal invariant such as extremal length. By the cover classification every connected Riemann surface, in particular each of $A_{r}$ and $D^{*}$ , is biholomorphic to a quotient of exactly one of the three models by a group of holomorphic automorphisms acting freely and properly discontinuously (Every Riemann surface is a quotient of a simply connected model).

Facts & Assumptions

Given: The Axiom of Choice; a real number $0 < r < 1$ and the sets $S_{r}$ , $A_{r}$ , $H_{-}$ , $D^{*}$ above (The Axiom of Choice, The natural logarithm as the inverse of the exponential function, Annuli in the complex plane, Riemann surfaces and holomorphic atlases).

[A1]

The Axiom of Choice (The Axiom of Choice) is used only through the universal-covering-type definition [F11] and the cover classification [F12], both of which assume it, and as Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) in the lifted structure [F21]; the remaining argument makes only finite or canonical choices.

[F1]

Kernel and fibres of the exponential ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ): $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ holds exactly when $z - w \in 2 π i Z$ .

[F2]

Cartesian form and modulus ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ): for real $x, y$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ and $∣ \exp (x + i y) ∣ = e^{x}$ .

[F3]

The real exponential is onto $(0, \infty)$ (The exponential is a continuous bijection from $R$ onto $(0, \infty)$ ): $\exp : R \to (0, \infty)$ is a bijection.

[F4]

Strict monotonicity (The exponential function is strictly increasing): $x \mapsto e^{x}$ is continuous and strictly increasing on $R$ .

[F5]

The real logarithm (The natural logarithm as the inverse of the exponential function): for $x > 0$ , $\log x$ is the unique real $y$ with $e^{y} = x$ , so $\log$ is the inverse function of the real exponential and both are increasing.

[F6]

The principal logarithm (The principal logarithm is a biholomorphism from the slit plane to the principal strip): the principal logarithm is a biholomorphism from the slit plane $C ∖ (- \infty, 0]$ onto the horizontal strip ${w : - π < Im w < π}$ , with inverse the exponential restricted to that strip; in particular $Log$ is holomorphic on the slit plane, $\exp (Log Z) = Z$ there, and $Log (\exp w) = w$ for $∣ Im w ∣ < π$ .

[F7]

The open mapping theorem (Open mapping theorem for holomorphic functions): every nonconstant holomorphic function on a complex domain is an open map.

[F8]

Covering maps and evenly covered neighbourhoods (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings): a covering map is a continuous surjection every point of whose base has an open neighbourhood $U$ with $p^{- 1} (U)$ a disjoint union of open sheets, each mapped homeomorphically onto $U$ by $p$ .

[F9]

Deck transformations (Deck transformations and the deck-transformation group of a covering): a deck transformation of $p : E \to B$ is an isomorphism $h : E \to E$ over $B$ , that is, a homeomorphism with $p \circ h = p$ , and the deck transformations form a group.

[F10]

Universal covering spaces (Universal covering spaces): a universal covering space of $B$ is a covering map $p : \tilde{B} \to B$ with $\tilde{B}$ simply connected.

[F11]

The universal-covering type (Spherical, parabolic and hyperbolic universal-covering types): the holomorphic universal cover of a connected Riemann surface is biholomorphic to exactly one of the Riemann sphere, the plane and the disc, the label is independent of the chosen cover, and the surface is hyperbolic precisely when its cover is biholomorphic to $D$ .

[F12]

Cover classification (Every Riemann surface is a quotient of a simply connected model): under the Axiom of Choice every connected Riemann surface is biholomorphic to the quotient of exactly one of the Riemann sphere, the complex plane and the unit disc by a group of holomorphic automorphisms acting freely and properly discontinuously.

[F13]

The three models (The sphere, plane and disc are pairwise biholomorphically distinct): $\hat{C}$ , $C$ and $D$ are simply connected Riemann surfaces and no two of them are biholomorphic.

[F14]

The Cayley map (Hyperbolic distances and geodesics in disc and half-plane): the map $C (ζ) = (ζ - i) / (ζ + i)$ maps the upper half-plane $H = {Im > 0}$ biholomorphically onto $D$ .

[F15]

Annuli (Annuli in the complex plane): $A (a; r, R) = {z : r < ∣ z - a ∣ < R}$ is the annulus about $a$ with inner radius $r$ and outer radius $R$ , and $A (a; 0, R)$ is the punctured disc $0 < ∣ z - a ∣ < R$ .

[F16]

Riemann surfaces (Riemann surfaces and holomorphic atlases): a Riemann surface is a nonempty connected Hausdorff second countable space with a holomorphic atlas, and every nonempty connected open subset of $C$ is one with the atlas of inclusions.

[F17]

The circle parametrization ( $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle): every point of the unit circle is $(\cos θ, \sin θ)$ for a unique $θ \in [0, 2 π)$ .

[F18]

Biholomorphisms (Biholomorphic maps between complex domains): a bijective holomorphic map with holomorphic inverse is a biholomorphism, and compositions and inverses of biholomorphisms are again biholomorphic.

[F19]

Fundamental groups of homeomorphic spaces (The fundamental group is a functor $π_{1} : {T o p}_{*} \to G r p$ ): a homeomorphism induces an isomorphism of fundamental groups, so simple connectivity is a topological property.

[F20]

Simply connected spaces (Simply connected topological spaces): a space is simply connected when it is nonempty and path connected and its fundamental group is trivial.

[F21]

The lifted holomorphic structure (A universal covering of a Riemann surface inherits a unique complex structure): under Countable Choice, a topological universal covering of a Riemann surface carries a unique complex structure making the projection a holomorphic unbranched covering, its total space is second countable, and its deck transformations are biholomorphic.

[F22]

Deck transformations are isometries (Deck transformations preserve the hyperbolic metric): for a disc uniformization of a hyperbolic surface, every deck transformation preserves the pulled-back Poincaré metric, its lengths and its distance, and the quotient metric is the surface Poincaré metric.

Proof technique: direct.

Verification

1.1F15F16given

Setup. $S_{r}$ and $H_{-}$ are open vertical strips and half-planes, hence convex and connected, and $A_{r} = A (0; r, 1)$ , $D^{*} = A (0; 0, 1)$ are the annuli of inner radius $r$ and $0$ ; all four sets are nonempty connected open subsets of $C$ , hence connected Riemann surfaces with the atlas of inclusions.

1.2F1

Injectivity on small discs. If $∣ w - w^{'} ∣ < 2 π$ and $\exp w = \exp w^{'}$ , then $w - w^{'} \in 2 π i Z$ [F1], so either $w = w^{'}$ or $∣ w - w^{'} ∣ \geq 2 π$ , a contradiction; hence $\exp$ is injective on every subset of diameter $< 2 π$ , in particular on every disc of radius at most $π$ .

1.3F6

The principal logarithm. The principal logarithm $Log$ is a biholomorphism from the slit plane $C ∖ (- \infty, 0]$ onto the horizontal strip $P = {w : ∣ Im w ∣ < π}$ , with inverse $\exp ∣_{P}$ ; thus $Log$ is holomorphic, $\exp (Log Z) = Z$ for $Z$ in the slit plane, and $Log (\exp w) = w$ whenever $∣ Im w ∣ < π$ [F6].

2.1F2F3F4F5step 1.1

The preimages of the two bases. For $w \in C$ one has $∣ \exp w ∣ = e^{Re w}$ [F2], so $\exp w \in A_{r}$ if and only if $r < e^{Re w} < 1$ , if and only if $\log r < Re w < 0$ , and $\exp w \in D^{*}$ if and only if $0 < e^{Re w} < 1$ , that is $Re w < 0$ ; the equivalences use that $x \mapsto e^{x}$ is strictly increasing with inverse $\log$ [F3, F4, F5]. Hence $\exp^{- 1} (A_{r}) = S_{r}$ and $\exp^{- 1} (D^{*}) = H_{-}$ .

2.2F1step 1.1

Translation invariance. For each $k \in Z$ the translation $τ_{k} (w) : = w + 2 π i k$ preserves real parts, hence maps $S_{r}$ onto $S_{r}$ and $H_{-}$ onto $H_{-}$ , with inverse $τ_{- k}$ ; and $\exp \circ τ_{k} = \exp$ on all of $C$ because $2 π i k \in \ker \exp$ [F1]. In particular $τ : = τ_{1}$ is the translation by $2 π i$ .

2.3F2F6F18step 1.1algebra

The strip is biholomorphic to the half-plane. Define $φ (w) : = \exp (i π (w - \log r) / L)$ on $C$ . For $w = x + i y \in S_{r}$ one has $φ (w) = e^{- π y / L} e^{i π (x - \log r) / L}$ [F2], where the modulus is positive and the argument $π (x - \log r) / L$ lies in $(0, π)$ , so $φ (w) \in H$ . Conversely, for $Z \in H$ put $ψ (Z) : = \log r - \frac{i L}{π} Log Z$ ; since $H$ is contained in the slit plane, $ψ$ is holomorphic [F6, F18], and writing $Log Z = u + i v$ with $v \in (- π, π)$ one has $Im Z = e^{u} \sin v > 0$ , so $v \in (0, π)$ and $ψ (Z) = \log r + \frac{L v}{π} - \frac{i L u}{π}$ has real part in $(\log r, 0)$ , that is $ψ (Z) \in S_{r}$ . The identity $\frac{i π (ψ (Z) - \log r)}{L} = Log Z$ gives $φ (ψ (Z)) = \exp (Log Z) = Z$ [F6], and for $w \in S_{r}$ the number $\frac{i π (w - \log r)}{L}$ lies in $P$ and exponentiates to $φ (w)$ , so it equals $Log φ (w)$ by the injectivity of $\exp ∣_{P}$ in [F6], and therefore $ψ (φ (w)) = w$ . Thus $φ : S_{r} \to H$ is a bijection, holomorphic with holomorphic inverse: a biholomorphism [F18].

3.1F2F3F5F17step 2.1

Both restrictions are onto. Let $z \in A_{r}$ ; by [F17] there is $θ \in [0, 2 π)$ with $z = ∣ z ∣ (\cos θ + i \sin θ)$ , and by [F3, F5] there is a unique real $x \in (\log r, 0)$ with $e^{x} = ∣ z ∣$ ; then $\exp (x + i θ) = e^{x} (\cos θ + i \sin θ) = z$ [F2], so $z \in \exp (S_{r})$ , and with step 2.1 the image of $S_{r}$ is exactly $A_{r}$ . The same argument with $x < 0$ and arbitrary $θ$ gives $\exp (H_{-}) = D^{*}$ .

3.2F14F18step 2.3

Biholomorphisms onto the disc. The map $w \mapsto - i w$ is a biholomorphism $H_{-} \to H$ with inverse $w \mapsto i w$ , since it is complex linear, bijective, and $Im (- i w) = - Re w > 0$ exactly when $w \in H_{-}$ [F18]; and by [F14] the Cayley map $C$ is a biholomorphism $H \to D$ . Hence $C \circ (- i \cdot)$ is a biholomorphism $H_{-} \to D$ and $C \circ φ$ is a biholomorphism $S_{r} \to D$ , compositions of biholomorphisms being biholomorphic [F18].

4.1F1F7step 2.1step 3.1step 2.2step 1.2

Evenly covered neighbourhoods over the annulus. Let $z_{0} \in A_{r}$ and, by step 3.1, choose $w_{0} \in S_{r}$ with $\exp w_{0} = z_{0}$ . Since $S_{r}$ is open and $w_{0} \in S_{r}$ , choose $0 < δ < π$ with $D (w_{0}, δ) \subseteq S_{r}$ , and put $U : = \exp (D (w_{0}, δ))$ . Then $U$ is open, because $\exp$ is a nonconstant holomorphic function on the domain $D (w_{0}, δ)$ [F7], and $z_{0} \in U \subseteq A_{r}$ by step 2.1. By [F1], $\exp^{- 1} (U) = ⋃_{k \in Z} (D (w_{0}, δ) + 2 π i k)$ : indeed $\exp w \in U$ holds exactly when $\exp w = \exp w^{'}$ for some $w^{'} \in D (w_{0}, δ)$ , that is, when $w - w^{'} \in 2 π i Z$ for such a $w^{'}$ . By step 2.2 every sheet $D (w_{0}, δ) + 2 π i k$ is contained in $S_{r}$ , and two distinct ones are disjoint, since an element of their intersection would exhibit $w_{1}, w_{2} \in D (w_{0}, δ)$ with $w_{1} - w_{2} = 2 π i (l - k) \neq 0$ of modulus $\geq 2 π$ , while $∣ w_{1} - w_{2} ∣ < 2 δ < 2 π$ . Finally, $\exp$ is injective on each sheet by step 1.2 and maps it onto $U$ , using $\exp (w^{'} + 2 π i k) = \exp w^{'}$ [F1]; hence $U$ is an evenly covered neighbourhood of $z_{0}$ with sheets $D (w_{0}, δ) + 2 π i k$ .

4.2F13F19F20step 3.2

Simple connectivity of the covering domains. The disc $D$ is simply connected [F13], hence nonempty, path connected and with trivial fundamental group [F20]; a biholomorphism is a homeomorphism, and a homeomorphism induces an isomorphism of fundamental groups [F19], so the biholomorphic images $S_{r}$ and $H_{-}$ are nonempty, path connected and have trivial fundamental group: they are simply connected [F19, F20, step 3.2].

5.1F1F7step 3.1step 2.2step 1.2

Evenly covered neighbourhoods over the punctured disc. Let $z_{0} \in D^{*}$ and choose $w_{0} \in H_{-}$ with $\exp w_{0} = z_{0}$ (step 3.1), and $0 < δ < \min (π, - Re w_{0})$ , so that $D (w_{0}, δ) \subseteq H_{-}$ ; the same computation as in step 4.1 shows that $U : = \exp (D (w_{0}, δ))$ is an open neighbourhood of $z_{0}$ contained in $D^{*}$ whose preimage is the disjoint union of the sheets $D (w_{0}, δ) + 2 π i k$ , each mapped homeomorphically onto $U$ by $\exp$ .

5.2F8step 2.1step 3.1step 4.1

The exponential over the annulus is a covering. The map $\exp : S_{r} \to A_{r}$ is continuous, its image is all of $A_{r}$ (step 3.1), and every $z_{0} \in A_{r}$ has the evenly covered neighbourhood produced in step 4.1; hence it is a covering map [F8], and $\exp^{- 1} (A_{r}) = S_{r}$ is step 2.1.

6.1F8step 2.1step 3.1step 5.1

The exponential over the punctured disc is a covering. The same argument with step 5.1 shows that $\exp : H_{-} \to D^{*}$ is a covering map, and $\exp^{- 1} (D^{*}) = H_{-}$ is step 2.1.

6.2F1F9step 1.1step 2.2

The deck group over the annulus. Since $\exp : S_{r} \to A_{r}$ is a covering map (step 5.2), its deck transformations are the homeomorphisms $h : S_{r} \to S_{r}$ with $\exp \circ h = \exp$ [F9]; let $h$ be one of them. For $w \in S_{r}$ the equality $\exp (h (w)) = \exp (w)$ gives $h (w) - w \in 2 π i Z$ [F1], and $w \mapsto h (w) - w$ is a continuous map from the connected strip $S_{r}$ into the discrete set $2 π i Z$ , hence is constant: $h (w) = w + 2 π i k$ for a fixed $k \in Z$ . Conversely every $τ_{k}$ is a homeomorphism of $S_{r}$ onto itself with $\exp \circ τ_{k} = \exp$ (step 2.2). Therefore $Deck (\exp ∣_{S_{r}}) = {τ_{k} : k \in Z}$ , an infinite cyclic group generated by $τ_{1}$ , since $τ_{k} \circ τ_{l} = τ_{k + l}$ and $τ_{1}^{k} = τ_{k}$ .

7.1F1F9step 1.1step 2.2

The deck group over the punctured disc. Since $\exp : H_{-} \to D^{*}$ is a covering map (step 6.1), the identical argument on the connected half-plane $H_{-}$ gives $Deck (\exp ∣_{H_{-}}) = {τ_{k} : k \in Z}$ , again infinite cyclic and generated by $τ_{1}$ .

7.2A1F10F11F12F13F21step 2.3step 3.2step 4.2step 5.2step 6.1

Universal covers and hyperbolic type. By steps 5.2, 6.1 and 4.2 the two exponential maps are covering maps with simply connected total space, hence universal covering spaces [F10]; the complex structures of $S_{r}$ and $H_{-}$ as open subsets of $C$ make $\exp$ holomorphic, so by the uniqueness of the lifted structure [F21] these are the holomorphic universal covers of $A_{r}$ and $D^{*}$ . The type definition [F11] then assigns to each of the two connected Riemann surfaces its unique type; since the exhibited covers are biholomorphic to $D$ (steps 2.3, 3.2) and no two of the three models are biholomorphic [F13], both $A_{r}$ and $D^{*}$ have hyperbolic universal-covering type. Under the Axiom of Choice [A1], the cover classification [F12] moreover exhibits each of the two surfaces as a quotient of $D$ by a group of holomorphic automorphisms acting freely and properly discontinuously, namely the conjugates of the deck groups of steps 6.2 and 7.1 under the uniformizations of steps 2.3 and 3.2.

8.1F22step 6.2step 7.1step 7.2∎

Conclusion. Steps 6.2 and 7.1 identify the two deck groups as the infinite cyclic groups generated by the translation $τ (w) = w + 2 π i$ , and step 7.2 identifies both base surfaces as hyperbolic; moreover, by [F22] the deck translations preserve the pulled-back Poincaré metric, its lengths and its distance of the corresponding uniformization. This proves all three assertions of the Example.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02Open item page →

A complex torus has a lattice of parabolic deck translations

Example

Assume the Axiom of Choice. Let $ω_{1}, ω_{2} \in C$ be $R$ -linearly independent, put $Λ : = Z ω_{1} + Z ω_{2}$ , and let $T : = C / Λ$ be the quotient of the translation action of $Λ$ on $C$ with quotient map $q$ . Then:

$q : C \to T$ is a covering map with simply connected total space, hence a universal covering space, and $Deck (q) = {z \mapsto z + λ : λ \in Λ}$ ; this deck group is isomorphic to $Λ$ and to $Z^{2}$ ;
$T$ is a compact Riemann surface, the complex torus of the lattice, for which $q$ is holomorphic;
$T$ has genus $1$ and parabolic universal-covering type.

Facts & Assumptions

Given: The Axiom of Choice; $R$ -linearly independent $ω_{1}, ω_{2} \in C$ ; the lattice $Λ = Z ω_{1} + Z ω_{2}$ ; the quotient space $T = C / Λ$ of the translation action with quotient map $q$ ; the standard torus $T^{2} = (R / Z)^{2}$ ; and the square schema $Y$ , the one-polygon schema with boundary word $a b a^{- 1} b^{- 1}$ .

[F1]

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function. In this example it is used only through the genus definition [F13] and the compact-genus corollary [F14], both of which assume it; every selection made below is finite or canonical.

[F2]

The coordinate and metric dictionary ( $C$ is the real coordinate plane, with coordinate arithmetic, The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane): the bijection $Φ (a + b i) = (a, b)$ carries addition and complex multiplication to the coordinatewise formulas of $R^{2}$ , so in particular it carries addition and real scalar multiplication to the coordinatewise operations, and $∣ z - w ∣ = ∥ Φ (z) - Φ (w) ∥_{2}$ ; hence the metric, convergence and continuity notions of $C$ are exactly their Euclidean counterparts, the metric topology is the usual topology of $R^{2}$ , and $C$ is a $2$ -dimensional real vector space.

[F3]

The field and modulus laws ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive): $C$ is a field, so addition is associative and commutative with identity $0$ and inverse $- λ$ , and $(z + λ) - (w + λ) = z - w$ ; and $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ , $∣ z ∣ \geq 0$ , $∣ z ∣ = 0$ exactly when $z = 0$ .

[F4]

Euclidean linear maps and continuity (A linear map $L : R^{m} \to R^{n}$ in Euclidean coordinates, Every Euclidean linear map has a unique matrix and satisfies $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ , Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction, Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form): for every real-linear $L : R^{m} \to R^{n}$ there is $K \geq 0$ with $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for all $h$ ; such an $L$ is Lipschitz with constant $K$ , hence uniformly continuous, hence continuous.

[F5]

Group actions and covering-space actions (Left group actions, transitive actions, and faithful actions, Covering-space actions by disjoint translates of neighbourhoods, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological): a left action of a group $G$ on a space $E$ satisfies $e \cdot x = x$ and $(g h) \cdot x = g \cdot (h \cdot x)$ ; it is an action by homeomorphisms when each $x \mapsto g \cdot x$ is a homeomorphism of $E$ ; and it is a covering-space action when every $e \in E$ has an open neighbourhood $U$ with $g U \cap U = \emptyset$ for every nonidentity $g \in G$ , in which case distinct translates of $U$ are disjoint.

[F6]

The orbit-map theorem (The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected): for a covering-space action of $G$ on $E$ the orbit map $E \to E / G$ is a covering, and if $E$ is path-connected then the deck group of this covering consists exactly of the transformations supplied by $G$ .

[F7]

Coverings, deck groups and universal covers (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Deck transformations and the deck-transformation group of a covering, Universal covering spaces): a covering map is a continuous surjection every point of whose base has an evenly covered neighbourhood; deck transformations are the isomorphisms over the base and form a group; a universal covering space is a covering whose total space is simply connected.

[F8]

Convexity and connected images (Every nonempty convex subset of $R^{n}$ is simply connected, Simply connected topological spaces, A continuous image of a connected space is connected, and connectedness is a topological property): every nonempty convex subset of $R^{n}$ is simply connected; a simply connected space is nonempty and path connected with trivial fundamental group; and a continuous image of a connected space is connected, so a continuous surjection from a connected space has connected codomain.

[F9]

The quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Characteristic properties: a map into a space with the initial topology is continuous iff every composite with the defining family is, a map out of a space with the final topology is continuous iff every composite with the defining family is, and the two topologies are respectively the coarsest and the finest making that family continuous): for a surjection $q : X \to Y$ , a subset $V \subseteq Y$ is open exactly when $q^{- 1} [V]$ is open in $X$ ; equivalently $Y$ carries the final topology of $q$ , so by the characteristic property a map $k : Y \to W$ is continuous exactly when $k \circ q$ is continuous.

[F10]

[F11]

Riemann surfaces and holomorphic translations (Riemann surfaces and holomorphic atlases, Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero): a Riemann surface is a nonempty connected Hausdorff second-countable space with a holomorphic atlas, whose charts are homeomorphisms onto open subsets of $C$ and whose pairwise transitions are holomorphic in both directions; complex polynomials are entire, so each translation $z \mapsto z + λ$ and each inverse $z \mapsto z - λ$ is holomorphic on $C$ .

[F12]

The square schema and the torus (Polygonal schemas and paired boundary edges, Torus commutator polygon, The two-dimensional torus $T^{2} = (R / Z)^{2}$ ): the one-polygon schema with boundary word $a b a^{- 1} b^{- 1}$ is a connected one-polygon schema whose realization $Y$ is a nonempty compact connected Hausdorff second-countable topological $2$ -manifold with one vertex class, two edge classes and one face, $Y$ carrying the quotient topology of the square by the side pairings; and that schema realizes the torus $T^{2} = (R / Z)^{2}$ .

[F13]

Genus by classification (Genus and Euler characteristic of a compact Riemann surface, Topological classification of compact Riemann surfaces): under the Axiom of Choice every compact Riemann surface is homeomorphic to $#_{g} T^{2}$ for exactly one $g \geq 0$ , where $#_{g} T^{2}$ is the connected sum of $g$ copies of the torus and $#_{0} T^{2} = S^{2}$ ; that number is the genus, and the one-fold connected sum $#_{1} T^{2}$ is the torus $T^{2}$ itself.

[F14]

Uniformization type (Spherical, parabolic and hyperbolic universal-covering types, The genus of a compact Riemann surface determines its uniformization type): a connected Riemann surface whose holomorphic universal cover is biholomorphic to $C$ is parabolic; and under the Axiom of Choice a compact Riemann surface of genus $1$ has parabolic type.

[F15]

Group isomorphisms (Group isomorphisms, automorphisms and the set $Aut (G)$ ): a bijective group homomorphism is a group isomorphism.

Proof technique: direct.

Verification

1.1F2F4givenalgebra

The two periods are a real basis. The map $T_{0} : R^{2} \to C$ , $T_{0} (s, t) : = s ω_{1} + t ω_{2}$ , is real-linear and injective: $s ω_{1} + t ω_{2} = 0$ forces $s = t = 0$ by $R$ -linear independence. Hence the composite $Φ \circ T_{0} : R^{2} \to R^{2}$ is an injective real-linear map of a $2$ -dimensional space into itself, so it is bijective and $T_{0}$ is a bijection [F2, F4]. Applying the boundedness bound of [F4] to the inverse linear map $(Φ \circ T_{0})^{- 1}$ gives $K \geq 0$ with $∥ (Φ \circ T_{0})^{- 1} v ∥_{2} \leq K ∥ v ∥_{2}$ for all $v$ ; here $K > 0$ , since $K = 0$ would make the inverse map zero, impossible for a bijection. Put $c : = 1 / K > 0$ . For $λ = m ω_{1} + n ω_{2} \in Λ$ one has $(m, n) = (Φ \circ T_{0})^{- 1} Φ (λ)$ , hence $\max (∣ m ∣, ∣ n ∣) \leq ∥ (m, n) ∥_{2} \leq K ∣ λ ∣$ , that is $∣ λ ∣ \geq c \max (∣ m ∣, ∣ n ∣)$ ; in particular $∣ λ ∣ \geq c$ for every nonzero $λ \in Λ$ .

1.2F2F8

The plane is simply connected. Under the dictionary of [F2] the space $C$ is the nonempty convex set $R^{2}$ , which is simply connected [F8]; in particular $C$ is nonempty and path connected with trivial fundamental group.

2.1F2F3F4F5step 1.1

Continuity and translations. The map $T_{0}$ is continuous: $Φ \circ T_{0}$ is real-linear, hence bounded and Lipschitz, hence continuous by [F4], and $Φ^{- 1}$ is an isometry by [F2]. For each $λ \in C$ the translation $τ_{λ} (z) : = z + λ$ satisfies $∣ τ_{λ} (z) - τ_{λ} (w) ∣ = ∣ (z + λ) - (w + λ) ∣ = ∣ z - w ∣$ for all $z, w$ by [F3], so $τ_{λ}$ is an isometry of $C$ ; it is therefore a bijection with continuous inverse $τ_{- λ}$ , that is, a homeomorphism of $C$ [F4, F5].

3.1F3F5step 1.1step 2.1

The translation action is a covering-space action. The set $Λ$ is a subgroup of $(C, +)$ [F3], and $λ \cdot z : = λ + z$ defines a left action of $Λ$ on $C$ by homeomorphisms: $(λ + μ) \cdot z = z + (λ + μ) = λ + (z + μ) = λ \cdot (μ \cdot z)$ and $0 \cdot z = z$ by the field laws [F3], while each map $z \mapsto λ \cdot z$ is the homeomorphism $τ_{λ}$ of step 2.1 [F5]. It is a covering-space action: fix $z \in C$ and put $U : = D (z, c / 4)$ ; if $w \in (U + λ) \cap U$ for some nonzero $λ \in Λ$ , then $w = u + λ = u^{'}$ for some $u, u^{'} \in U$ , so $λ = u^{'} - u$ and $∣ λ ∣ \leq ∣ u^{'} - z ∣ + ∣ z - u ∣ < c / 2 < c$ , contradicting $∣ λ ∣ \geq c$ from step 1.1; hence $(U + λ) \cap U = \emptyset$ for every nonzero $λ$ , as required by [F5].

3.2F5F9step 2.1

The quotient map is open. For open $W \subseteq C$ one has $q^{- 1} (q (W)) = ⋃_{λ \in Λ} (W + λ)$ , because the classes of $q$ are the orbits ${λ + z : λ \in Λ}$ ; each $W + λ = τ_{λ} (W)$ is open by step 2.1, so $q^{- 1} (q (W))$ is open in $C$ and $q (W)$ is open in $T$ by the quotient topology [F9]. Thus $q$ is an open map.

4.1F6F7F9step 3.1step 1.2

The orbit map is a covering with deck group the translations. The space $T = C / Λ$ is the orbit space of the action of step 3.1 and $q$ is its orbit map [F9]. By steps 3.1 and 1.2 the orbit-map theorem [F6] applies: $q : C \to T$ is a covering map, and since $C$ is path connected its deck group consists exactly of the transformations supplied by $Λ$ , that is, $Deck (q) = {τ_{λ} : λ \in Λ}$ . Since $C$ is simply connected (step 1.2), $q$ is a universal covering space [F7].

4.2F5F9F11step 1.1step 3.2

Small discs give charts. Fix $z \in C$ and put $D_{z} : = D (z, c / 3)$ and $U_{z} : = q (D_{z})$ ; the set $U_{z}$ is open in $T$ by step 3.2. If $q (w) = q (w^{'})$ with $w, w^{'} \in D_{z}$ , then $w^{'} = λ + w$ for some $λ \in Λ$ , because the classes of the orbit map are the orbits [F5, F9]; then $w - w^{'} \in Λ$ and $∣ w - w^{'} ∣ < 2 c / 3 < c$ , so $w = w^{'}$ by step 1.1. Hence $q_{z} : = q ∣_{D_{z}}$ is a bijection $D_{z} \to U_{z}$ , and it is an open continuous map: for open $A \subseteq D_{z}$ the set $q (A)$ is open in $T$ by step 3.2, hence open in $U_{z}$ . Therefore its inverse $φ_{z} : = q_{z}^{- 1} : U_{z} \to D_{z}$ is a homeomorphism onto the open set $D_{z} \subseteq C$ , that is, a chart [F5, F11]. The sets $U_{z}$ cover $T$ , because $q$ is onto and $z \in D_{z}$ for every $z$ .

4.3F3F9step 1.1step 3.2

The quotient is Hausdorff. Let $[z] \neq [z^{'}]$ in $T$ and put $P : = z - z^{'} \notin Λ$ . With $R : = 2 ∣ P ∣ + 1$ , the set $S : = Λ \cap \overline{D} (0, R)$ is finite: by step 1.1 every $λ = m ω_{1} + n ω_{2} \in S$ has $\max (∣ m ∣, ∣ n ∣) \leq R / c$ , and only finitely many integer pairs satisfy this. Since $0 \in S$ , the number $δ : = dist (P, S) = \min {∣ P - λ ∣ : λ \in S}$ is positive and $δ \leq ∣ P ∣$ ; and for $λ \in Λ ∖ S$ one has $∣ P - λ ∣ \geq ∣ λ ∣ - ∣ P ∣ > R - ∣ P ∣ = ∣ P ∣ + 1 > δ$ by [F3]. Hence $dist (P, Λ) = δ > 0$ . The open sets $q (D (z, δ / 2))$ and $q (D (z^{'}, δ / 2))$ are then disjoint: a common class would give $u \in D (z, δ / 2)$ and $u^{'} \in D (z^{'}, δ / 2)$ with $u - u^{'} \in Λ$ , whence $∣ P - (u - u^{'}) ∣ \leq ∣ z - u ∣ + ∣ u^{'} - z^{'} ∣ < δ$ , contradicting $dist (P, Λ) = δ$ ; both sets are open by step 3.2. Therefore $T$ is Hausdorff.

5.1F3F15step 1.1step 4.1

The deck group is $Z^{2}$ . By step 4.1 the map $λ \mapsto τ_{λ}$ is a bijection $Λ \to Deck (q)$ and $τ_{λ} \circ τ_{μ} = τ_{λ + μ}$ , $τ_{0} = {id}_{C}$ , so it is a group isomorphism [F15]. The map $Z^{2} \to Λ$ , $(m, n) \mapsto m ω_{1} + n ω_{2}$ , is surjective by the definition of $Λ$ and injective because $m ω_{1} + n ω_{2} = 0$ forces $m = n = 0$ , and it is additive; hence it too is an isomorphism [F15]. Therefore $Deck (q) ≅ Λ ≅ Z^{2}$ .

5.2F11F3step 1.1step 2.1step 4.2

Transitions are translations. Let $z, z^{'} \in C$ with $W : = U_{z} \cap U_{z^{'}} \neq \emptyset$ . For $u \in φ_{z} (W) \subseteq D_{z}$ the point $φ_{z^{'}} (q (u))$ lies in $D_{z^{'}}$ and satisfies $q (φ_{z^{'}} (q (u))) = q (u)$ , since $q (u) \in W \subseteq U_{z^{'}}$ and $φ_{z^{'}}$ inverts $q$ on $D_{z^{'}}$ (step 4.2); hence $λ (u) : = φ_{z^{'}} (q (u)) - u \in Λ$ by the orbit description of the classes. The map $u \mapsto λ (u)$ is continuous on the open set $φ_{z} (W)$ (compositions of continuous maps and subtraction, steps 2.1 and 4.2) and its values are separated: $∣ λ - μ ∣ \geq c$ for distinct $λ, μ \in Λ$ (step 1.1). Given $u$ in the domain, continuity gives $δ > 0$ with $∣ λ (u^{'}) - λ (u) ∣ < c$ whenever $∣ u^{'} - u ∣ < δ$ ; two distinct values of $λ$ would differ by at least $c$ , so $λ$ is constant on $φ_{z} (W) \cap D (u, δ)$ . Thus the transition $φ_{z^{'}} \circ φ_{z}^{- 1}$ , which on $φ_{z} (W)$ is the map $u \mapsto u + λ (u)$ , agrees near each of its points with a single translation $u \mapsto u + λ_{0}$ , an entire function [F11]; the same argument with $z$ and $z^{'}$ interchanged shows that the inverse transition $φ_{z} \circ φ_{z^{'}}^{- 1}$ is holomorphic too. Hence the charts $φ_{z}$ are pairwise compatible.

5.3F5F9F10F12step 1.1step 2.1step 4.3

The square schema realizes the quotient. Let $g : [0, 1]^{2} \to T$ be $g (s, t) : = q (s ω_{1} + t ω_{2})$ , continuous as the composite of the continuous map $T_{0}$ (step 2.1) with $q$ [F9]. It respects the side pairings: $g (1, t) = q (ω_{1} + t ω_{2}) = q (t ω_{2}) = g (0, t)$ and $g (s, 1) = q (s ω_{1} + ω_{2}) = q (s ω_{1}) = g (s, 0)$ , since $ω_{1}, ω_{2} \in Λ$ and the classes of $q$ are the orbits [F5]. By the characteristic property of the quotient $Y$ of the square by these pairings [F9, F12], $g$ induces a continuous map $\overset{ˉ}{g} : Y \to T$ with $\overset{ˉ}{g} \circ π = g$ , where $π$ is the quotient map of the schema. The map $\overset{ˉ}{g}$ is surjective: given $[z] \in T$ write $z = T_{0} (s, t)$ (step 1.1), decompose $s = m + s^{'}$ , $t = n + t^{'}$ with $m, n \in Z$ and $s^{'}, t^{'} \in [0, 1)$ , and use $T_{0} (s, t) - T_{0} (s^{'}, t^{'}) = T_{0} (m, n) \in Λ$ to get $[z] = g (s^{'}, t^{'})$ . It is injective: if $g (s, t) = g (s^{'}, t^{'})$ then $T_{0} (s - s^{'}, t - t^{'}) \in Λ = T_{0} (Z^{2})$ , so $(s - s^{'}, t - t^{'}) \in Z^{2}$ by the injectivity of $T_{0}$ (step 1.1); since all four coordinates lie in $[0, 1]$ , the differences $s - s^{'}$ and $t - t^{'}$ lie in ${- 1, 0, 1}$ and each is nonzero exactly when the two points lie on a paired pair of sides, so $(s, t)$ and $(s^{'}, t^{'})$ have the same image under $π$ . Hence $\overset{ˉ}{g}$ is a continuous bijection; since $Y$ is compact [F12] and $T$ is Hausdorff (step 4.3), $\overset{ˉ}{g}$ is a homeomorphism [F10].

6.1F8F11F12step 1.2step 4.2step 5.2step 4.3step 5.3

The quotient is a compact Riemann surface. The charts ${φ_{z}}_{z \in C}$ cover $T$ and have holomorphic transitions in both directions (steps 4.2 and 5.2), so they form a holomorphic atlas; the space $T$ is nonempty, connected as the image of the connected space $C$ under the continuous surjection $q$ [F8, step 1.2], Hausdorff by step 4.3, and second countable because it is homeomorphic to $Y$ (step 5.3) and $Y$ is second countable [F12]. Therefore $T$ is a Riemann surface by [F11], and it is compact because $Y$ is compact [F12] and homeomorphic to $T$ (step 5.3). The map $q$ is holomorphic for this atlas: on $D_{z}$ the chart expression $φ_{z} \circ q$ is the identity, because $φ_{z}$ inverts $q ∣_{D_{z}}$ (step 4.2).

7.1F13step 5.3step 6.1

The genus is one. By [F12] the same square schema realizes the torus $T^{2}$ , so $Y ≅ T^{2}$ , and with step 5.3 this gives $T ≅ T^{2} = #_{1} T^{2}$ [F13]. The topological classification of compact Riemann surfaces [F13] supplies exactly one $g \geq 0$ with $T ≅ #_{g} T^{2}$ ; since $g = 1$ has this property, the genus of the compact Riemann surface $T$ is $1$ .

8.1F1F14step 1.2step 6.1step 7.1∎

The type is parabolic. By steps 6.1 and 7.1 the space $T$ is a compact Riemann surface of genus $1$ , so the compact-genus corollary [F14] gives $T$ parabolic universal-covering type under the Axiom of Choice [F1]. Moreover the exhibited covering $q : C \to T$ is holomorphic (step 6.1) with simply connected total space $C$ (step 1.2), hence is a holomorphic universal cover of $T$ whose model is $C$ , in agreement with the definition of parabolic type [F14]. This proves all three assertions of the Example.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

A genus-two compact surface gives a cocompact Fuchsian group

Example

Assume the Axiom of Choice. Let $a_{1}, \dots, a_{6} \in C$ be distinct, put $P (x) = \prod_{j = 1}^{6} (x - a_{j})$ and let $X_{0} = {(x, y) \in C^{2} : y^{2} = P (x)}$ be the affine hyperelliptic curve, completed at infinity by the two charts constructed in step 1.2. Then the completed space $X$ is a compact connected Riemann surface, and the projection $π : X \to \hat{C}, (x, y) \mapsto x, \infty_{\pm} \mapsto \infty,$ is a proper holomorphic map of degree $2$ whose branch values are exactly the six numbers $a_{1}, \dots, a_{6}$ , each carrying a single point of ramification index $2$ , while the value $\infty$ is unramified. Consequently:

$X$ is a compact Riemann surface of genus $2$ ;
$X$ has hyperbolic universal-covering type: its holomorphic universal cover $p : \tilde{X} \to X$ is biholomorphic to $D$ ;
writing $ψ : \tilde{X} \to D$ for a biholomorphism and $Γ : = ψ Deck (p) ψ^{- 1} \leq Aut (D)$ , the transported map $Ψ : = p \circ ψ^{- 1} : D \to X$ is a covering whose deck group is exactly $Γ$ ; the group $Γ$ is torsion-free, and it is discrete for the compact-open topology on $Aut (D)$ ; it acts on $D$ freely and properly discontinuously, and $D / Γ$ is homeomorphic to $X$ ;
$X$ is a cocompact Fuchsian quotient.

Here a subgroup of $Aut (D)$ is called Fuchsian when it acts on $D$ freely and properly discontinuously; its quotient is cocompact when that quotient is compact. No examples-page item and no Gauss-Bonnet theorem is consumed.

Facts & Assumptions

Given: The Axiom of Choice; distinct $a_{1}, \dots, a_{6} \in C$ ; $P (x) = \prod_{j = 1}^{6} (x - a_{j})$ and $u_{j} (x) = P (x) / (x - a_{j})$ ; the affine curve $X_{0} = {(x, y) \in C^{2} : y^{2} = P (x)}$ with the projection $π_{0} (x, y) = x$ ; the Riemann sphere $\hat{C}$ with its two standard charts; $Q (t) = \prod_{j = 1}^{6} (1 - a_{j} t)$ ; the completed space $X = X_{0} \cup {\infty_{+}, \infty_{-}}$ with the two charts at infinity of step 2.1 and the projection $π$ equal to $π_{0}$ on $X_{0}$ and sending $\infty_{\pm}$ to $\infty$ ; and a choice of $ε > 0$ with $Q \neq 0$ on $∣ t ∣ < ε$ together with a holomorphic square root $ρ$ of $Q$ on that disc.

[A1]

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function. It is used only through the genus interface [F13] and the existence and type assertions of the holomorphic universal cover [F14], both of which assume it; every selection made in the construction below is finite or explicit.

[F1]

Riemann surfaces, holomorphic maps and biholomorphisms (Riemann surfaces and holomorphic atlases, Holomorphic maps and meromorphic functions on Riemann surfaces, Biholomorphic maps between complex domains): a Riemann surface is a nonempty connected Hausdorff second-countable space with a holomorphic atlas whose charts are homeomorphisms onto plane domains and whose transitions are holomorphic in both directions; a map of Riemann surfaces is holomorphic when its chart expressions are holomorphic; a biholomorphism is a bijective holomorphic map with holomorphic inverse, and it is in particular a homeomorphism.

[F2]

The holomorphic implicit function theorem (The holomorphic implicit function theorem): if $F$ is holomorphic near $(a, b)$ , $F (a, b) = 0$ , and the partial derivative of $F$ in the second variable does not vanish at $(a, b)$ , then near $(a, b)$ the zero set of $F$ is the graph $w = φ (z)$ of a unique holomorphic $φ$ , and $F (z, w) = 0$ exactly when $w = φ (z)$ ; the same statement holds with the roles of the variables exchanged.

[F3]

Local logarithm and square root (A nonvanishing holomorphic function on a disc has a holomorphic logarithm): a nowhere-vanishing holomorphic function $h$ on a disc has a holomorphic logarithm $L$ with $\exp L = h$ ; then $ρ : = \exp (L / 2)$ is holomorphic with $ρ^{2} = h$ .

[F4]

The Riemann sphere and its charts (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity): the standard charts are $ϕ_{0} (z) = z$ on $\hat{C} ∖ {\infty}$ and $ϕ_{\infty}$ with $ϕ_{\infty} (z) = 1 / z$ for $z \in C^{\times}$ and $ϕ_{\infty} (\infty) = 0$ ; on the overlap the transition is $w \mapsto 1 / w$ , holomorphic on $C^{\times}$ .

[F5]

The sphere as a topological sphere (Stereographic projection identifies the Riemann sphere with the unit two-sphere): stereographic projection $Σ : \hat{C} \to S^{2}$ , with $S^{2} = {(x, y, t) \in R^{3} : x^{2} + y^{2} + t^{2} = 1}$ , is a homeomorphism.

[F6]

The sphere is connected ( $S^{n}$ is simply connected for every $n \geq 2$ , Simply connected topological spaces): $S^{2}$ is simply connected, hence nonempty and path connected, hence connected.

[F7]

Compactness (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones): a subset of $R^{n}$ is compact exactly when it is closed and bounded; continuous images of compact sets are compact; a closed subset of a compact space is compact; compact subsets of a Hausdorff space are closed and compact subsets admit finite ambient open subcovers.

[F8]

Connectedness tools (If $A$ is connected and $A \subseteq B \subseteq \overline{A}$ then $B$ is connected; in particular the closure of a connected set is connected, A continuous image of a connected space is connected, and connectedness is a topological property, A subset of $R$ is connected if and only if it is order-convex, that is, an interval, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets): a set squeezed between a connected set and its closure is connected, so closures of connected sets are connected; continuous images of connected sets are connected; a subset of $R$ is connected exactly when it is order-convex, so $[0, 2 π]$ is connected; and a two-point space such as ${1, - 1}$ is disconnected, being the union of its two nonempty open singletons.

[F9]

Path connectivity (The exterior of a closed disc in the plane is path-connected, Every path-connected space is connected, and every path component lies inside a component): for every real $R \geq 0$ and centre $c$ the exterior ${z \in C : ∣ z - c ∣ > R}$ is path connected, hence connected; and a path-connected space is connected.

[F10]

Ramification (Local power-map normal form on Riemann surfaces, Ramification index, ramification order and branch value): for a nonconstant holomorphic map of Riemann surfaces there are centred charts with expression $z \mapsto z^{e}$ , the exponent $e = e_{x} (f)$ is the ramification index, the index equals the order ${ord}_{x} (f - f (x))$ of the centred expression in any charts, and $e_{x} (f) = 1$ exactly when $f$ is a local biholomorphism at $x$ .

[F11]

Degree of a proper map (Degree of a proper holomorphic map of Riemann surfaces): for a proper nonconstant holomorphic map $f$ between connected Riemann surfaces the weighted fibre count $d = \sum_{x \in f^{- 1} (y)} e_{x} (f)$ is a positive finite integer independent of $y$ .

[F12]

Riemann-Hurwitz (Riemann–Hurwitz formula for compact Riemann surfaces): for a nonconstant holomorphic map $f : X \to Y$ of compact connected Riemann surfaces, $2 g (X) - 2 = d (2 g (Y) - 2) + \sum_{x \in X} (e_{x} - 1) .$

[F13]

Genus and classification (Genus and Euler characteristic of a compact Riemann surface, Topological classification of compact Riemann surfaces): under the Axiom of Choice every compact Riemann surface is homeomorphic to $#_{g} T^{2}$ for exactly one $g \geq 0$ , where $#_{0} T^{2} = S^{2}$ ; that number is the genus, and $g = 0$ holds exactly for the sphere.

[F14]

Universal cover, type and the compact-genus corollary (Spherical, parabolic and hyperbolic universal-covering types, The genus of a compact Riemann surface determines its uniformization type): under the Axiom of Choice a connected Riemann surface has a holomorphic universal cover $p : \tilde{X} \to X$ , every deck transformation is biholomorphic, and $\tilde{X}$ is biholomorphic to exactly one of $\hat{C}$ , $C$ , $D$ , the occurring model being the universal-covering type (spherical, parabolic, hyperbolic); and a compact Riemann surface of genus at least $2$ has hyperbolic type, so its holomorphic universal cover is biholomorphic to $D$ .

[F15]

Free and properly discontinuous actions (Free and properly discontinuous group actions): an action of a group $G$ on a space $Y$ by homeomorphisms is free when no nonidentity element fixes a point, and properly discontinuous when for every compact $K \subseteq Y$ only finitely many $g \in G$ satisfy $g K \cap K \neq \emptyset$ .

[F16]

Coverings, sheets and deck transformations (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Universal covering spaces, On a connected covering space, a deck transformation is determined by one point and the deck action is free): a covering map is a continuous surjection every point of whose base has an evenly covered neighbourhood $U$ whose preimage is a disjoint union of open sheets each mapped homeomorphically onto $U$ ; a universal covering is a covering with simply connected total space; deck transformations are the homeomorphisms over the base and form a group acting by evaluation; and for a covering with connected total space two deck transformations agreeing at one point are equal, so the deck group acts freely.

[F17]

The uniformization interface (Deck transformations preserve the hyperbolic metric): for a connected Riemann surface of hyperbolic universal-covering type with a uniformization $(p, ψ)$ , where $p : \tilde{X} \to X$ is its holomorphic universal covering and $ψ : \tilde{X} \to D$ is a biholomorphism, every $h \in Deck (p)$ is a biholomorphism of $\tilde{X}$ and the conjugate $γ_{h} : = ψ \circ h \circ ψ^{- 1}$ is an automorphism of $D$ .

[F18]

Deck transitivity on fibres (For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group): for a path-connected, locally path-connected, semilocally simply connected base the deck group of a universal cover is isomorphic to the fundamental group, the isomorphism carrying a loop class to the deck transformation that moves the chosen point of the fibre to the corresponding lifted endpoint; consequently the deck group acts transitively on every fibre.

[F19]

Disc automorphisms (Every automorphism of the disc is a rotated Blaschke factor, The unit disc, the upper half-plane, and Blaschke factors): a holomorphic map $f : D \to D$ is an automorphism of $D$ if and only if there are $a \in D$ and $θ \in R$ with $f (z) = e^{i θ} φ_{a} (z) = e^{i θ} \frac{a - z}{1 - \overline{a} z} (z \in D) .$

[F20]

Mobius transformations (Möbius transformations of the Riemann sphere, Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant): a Mobius transformation is a map $z \mapsto (a z + b) / (c z + d)$ with $a d - b c \neq 0$ , extended to $\hat{C}$ ; a nonidentity Mobius transformation is either parabolic, with one fixed point, and then conjugate to $z \mapsto z + 1$ , or has two fixed points and is conjugate to $z \mapsto λ z$ for some $λ \in C^{\times} ∖ {1}$ ; and for any representing matrix $A$ the quantity $τ (M) = tr (A)^{2} / \det A$ is independent of the representative, equals $λ + 2 + λ^{- 1}$ in the dilation normal form and equals $4$ in the translation normal form.

[F21]

The compact-open topology (The compact-open topology on $C (X, Y)$ for a metric domain $X$ , with subbasis $S (K, V) = {f : f [K] \subseteq V}$ , For a metric domain and a metric target the compact-open topology on $C (X, Y)$ is the topology of compact convergence, The topology of compact convergence on $C (X, Y)$ for metric $X$ and $Y$ : uniform convergence on each compact subset of $X$ ): on the set $C (D, D)$ of continuous maps the compact-open topology, generated by the sets $S (K, V) = {f : f [K] \subseteq V}$ over compact $K$ and open $V$ , is the same topology as the topology of compact convergence, for which the sets $B_{L} (g, δ) = {f : d (f (x), g (x)) < δ$ for every $x \in L}$ over compact $L \subseteq D$ and $δ > 0$ form a neighbourhood base at $g$ .

[F22]

Quotient topology and homeomorphisms (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Characteristic properties: a map into a space with the initial topology is continuous iff every composite with the defining family is, a map out of a space with the final topology is continuous iff every composite with the defining family is, and the two topologies are respectively the coarsest and the finest making that family continuous, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological): for a surjection $q : Y \to Z$ the quotient topology on $Z$ is the final topology of $q$ , so a subset of $Z$ is open exactly when its preimage under $q$ is open in $Y$ and a map $k : Z \to W$ is continuous exactly when $k \circ q$ is continuous; a continuous bijection whose inverse is continuous, equivalently a continuous bijection which is an open map, is a homeomorphism.

Proof technique: direct.

Verification

1.1F1F2givenalgebra

The affine curve and its local parameters. Let $F (x, y) : = y^{2} - P (x)$ and let $(x_{0}, y_{0}) \in X_{0}$ . The gradient $(- P^{'} (x), 2 y)$ does not vanish at $(x_{0}, y_{0})$ : if $y_{0} \neq 0$ this is read off the second component, while if $y_{0} = 0$ then $P (x_{0}) = 0$ , so $x_{0} = a_{j}$ for a unique $j$ and $- P^{'} (a_{j}) = - \prod_{k \neq j} (a_{j} - a_{k}) \neq 0$ because the six roots are distinct. By [F2] the curve is therefore locally a graph over a coordinate: where $y_{0} \neq 0$ there are discs $A ∋ x_{0}$ , $B ∋ y_{0}$ and a holomorphic $η : A \to B$ with $X_{0} \cap (A \times B) = {(x, η (x)) : x \in A}$ , and at $(a_{j}, 0)$ there are discs $A_{j} ∋ a_{j}$ , $B_{j} ∋ 0$ and a holomorphic $ψ_{j} : B_{j} \to A_{j}$ with $X_{0} \cap (A_{j} \times B_{j}) = {(ψ_{j} (y), y) : y \in B_{j}}$ . The maps $x \mapsto (x, η (x))$ and $y \mapsto (ψ_{j} (y), y)$ are homeomorphisms onto their images, with inverses given by the holomorphic coordinate functions $x$ and $y$ , so the two families of charts are compatible in both directions [F1]. Moreover $u_{j}$ is holomorphic near $a_{j}$ with $u_{j} (a_{j}) = P^{'} (a_{j}) \neq 0$ , so from $y^{2} = P (ψ_{j} (y)) = (ψ_{j} (y) - a_{j}) u_{j} (ψ_{j} (y))$ one obtains $ψ_{j} (y) - a_{j} = y^{2} w_{j} (y)$ with $w_{j} : = 1 / (u_{j} \circ ψ_{j})$ holomorphic near $0$ and $w_{j} (0) = 1 / P^{'} (a_{j}) \neq 0$ . Hence the projection $π_{0} (x, y) = x$ has, in the chart with parameter $y$ at $(a_{j}, 0)$ and the centred target chart $z \mapsto a_{j} + z$ , the expression $y \mapsto y^{2} w_{j} (y)$ , of order $2$ at $0$ , while at a point with $y \neq 0$ its expression in the chart with parameter $x$ is $z \mapsto z$ .

1.2F9given

The complement of the six roots is path connected. Let $V : = C ∖ {a_{1}, \dots, a_{6}}$ and fix $R > \max {1 / ε, 1 + \max_{j} ∣ a_{j} ∣}$ . For $x \in V$ with $∣ x ∣ \leq R$ , choose a direction $θ$ different from the at most six directions toward the roots. The segment of length $3 R$ from $x$ in that direction avoids the roots and ends in $E : = {z \in C : ∣ z ∣ > R}$ , since its endpoint has modulus at least $3 R - ∣ x ∣ \geq 2 R > R$ . If $∣ x ∣ > R$ , then $x \in E$ already. The set $E$ is path connected by [F9] and is contained in $V$ , so any two points of $V$ can be joined in $V$ by paths through $E$ . Hence $V$ is path connected and connected.

2.1F3F4givenalgebra

Completion at infinity. Put $t : = 1 / x$ and $v : = y / x^{3}$ , so that on $X_{0}$ with $x \neq 0$ the equation $y^{2} = P (x)$ reads $v^{2} = P (x) / x^{6} = Q (t) = \prod_{j = 1}^{6} (1 - a_{j} t)$ , and $Q (0) = 1$ . Choose $ε > 0$ with $Q (t) \neq 0$ for $∣ t ∣ < ε$ ; by [F3] there is a holomorphic $L$ on that disc with $\exp L = Q$ , and $ρ : = \exp (L / 2)$ satisfies $ρ^{2} = Q$ and $ρ (0) \neq 0$ . Add two points $\infty_{+}$ and $\infty_{-}$ to $X_{0}$ and declare $t \mapsto (t, + ρ (t)), t \mapsto (t, - ρ (t)) (∣ t ∣ < ε),$ to be charts at them. For $t \neq 0$ the corresponding point of the first chart is $(x, y) = (1 / t, ρ (t) t^{- 3})$ , and $y^{2} = t^{- 6} ρ (t)^{2} = t^{- 6} Q (t) = P (1 / t) = P (x)$ , so it lies in $X_{0}$ ; the two charts are glued to the affine charts of step 1.1 by the transition maps $t = 1 / x$ , $v = y / x^{3}$ and their inverses $x = 1 / t$ , $y = v / t^{3}$ , holomorphic on $t \neq 0$ . Thus $X : = X_{0} \cup {\infty_{+}, \infty_{-}}$ carries the atlas of steps 1.1 and 2.1, and towards the chart $ϕ_{\infty}$ of [F4] the projection $π$ has at $\infty_{\pm}$ the expression $t \mapsto t$ . Moreover for every $R > 1 / ε$ the points of $X$ with $∣ x ∣ > R$ , equivalently with $t = 1 / x$ satisfying $0 < ∣ t ∣ < 1 / R$ , are exactly the points of the two chart images with $t \neq 0$ , and their second coordinates are $\pm x^{3} ρ (1 / x)$ .

2.2F3F8F16step 1.1step 1.2

The punctured affine curve is a connected two-sheeted cover of $V$ . Put $Z : = {(x, y) \in X_{0} : x \in V}$ . For $x \in V$ one has $P (x) \neq 0$ , so by [F3] there are a disc $D (x, r) \subseteq V$ and a holomorphic square root $ρ_{x}$ of $P$ on it; the two maps $z \mapsto (z, \pm ρ_{x} (z))$ are local inverses of $π_{0}$ , and they exhibit $π_{0}^{- 1} (D (x, r))$ as the disjoint union of two open sets each mapped homeomorphically onto $D (x, r)$ . Hence $π_{0} : Z \to V$ is a covering of degree $2$ [F16], so it is continuous and open, and $Z$ is Hausdorff and second countable as a subspace of $C^{2}$ . Suppose $Z$ were disconnected, say $Z = Z_{1} ⊔ Z_{2}$ with the $Z_{i}$ nonempty, open and closed. Over each small evenly covered disc, each of the two connected sheets lies wholly in one of the clopen $Z_{i}$ , so the number of sheet points in $Z_{i}$ is locally constant on $V$ . Thus the images $π_{0} (Z_{i})$ are nonempty, open and closed in the connected space $V$ (step 1.2), hence equal to $V$ ; and since each fibre of $π_{0}$ consists of exactly two points, one over each $Z_{i}$ , the restriction $π_{0} ∣_{Z_{1}}$ is a bijection onto $V$ with local continuous inverse, hence a homeomorphism. Its inverse provides a continuous $s : V \to C$ with $s (x)^{2} = P (x)$ for all $x \in V$ . Fix $0 < ε_{1} < \min_{k \neq 1} ∣ a_{1} - a_{k} ∣$ and, by [F3], a holomorphic $ρ$ on $D (a_{1}, ε_{1})$ with $ρ^{2} = u_{1}$ ; there $u_{1}$ is holomorphic and nowhere zero. Define $c (θ) : = a_{1} + (ε_{1} / 2) e^{i θ}$ , $w (θ) : = (ε_{1} / 2)^{1 / 2} e^{i θ / 2}$ , so $w (θ)^{2} = c (θ) - a_{1}$ , and $g (θ) : = s (c (θ)) / (w (θ) ρ (c (θ)))$ for $θ \in [0, 2 π]$ . Then $g (θ)^{2} = \frac{P (c (θ))}{(c (θ) - a_{1}) u_{1} (c (θ))} = 1,$ so $g$ maps the connected interval $[0, 2 π]$ continuously into ${1, - 1}$ [F8]; a continuous image of a connected set is connected while ${1, - 1}$ is disconnected, so $g$ is constant [F8]. But $s (c (2 π)) = s (c (0))$ , $ρ (c (2 π)) = ρ (c (0))$ and $w (2 π) = - w (0)$ , so $g (2 π) = - g (0) \neq g (0)$ , a contradiction. Hence $Z$ is connected.

3.1F1step 1.1step 2.1

The completed space is Hausdorff and second countable, and $π$ is holomorphic for the atlas. Distinct points of $X_{0}$ are separated by the Hausdorff topology of $C^{2}$ , the affine charts being restrictions of the coordinate projections; the two points $\infty_{+}$ and $\infty_{-}$ are separated because their chart values at $t = 0$ are $ρ (0)$ and $- ρ (0)$ , which are distinct and give disjoint chart images. A point $(x_{0}, y_{0}) \in X_{0}$ with $R > \max {∣ x_{0} ∣, 1 / ε}$ and an infinity point are separated by the open sets ${∣ x ∣ < R} \cap X_{0}$ and the image under the relevant chart of the disc $∣ t ∣ < 1 / R$ , which is disjoint from the first by step 2.1. Hence $X$ is Hausdorff. A countable base of the topology of $X$ is obtained from a countable base of the open subspace $X_{0}$ , which is second countable as a subspace of $C^{2}$ , together with the images under the two chart maps of a countable base of the disc $∣ t ∣ < ε$ ; these sets are open and every open subset of $X$ is the union of its intersections with the three open pieces $X_{0}$ and the two chart images, so $X$ is second countable. The chart expressions of $π$ are holomorphic: $z \mapsto z$ and $y \mapsto ψ_{j} (y)$ on the affine charts of step 1.1 and $t \mapsto t$ on the two charts at infinity of step 2.1. Consequently, once $X$ is known to be connected, it is a Riemann surface with this atlas and $π$ is a nonconstant holomorphic map of Riemann surfaces [F1].

3.2F7step 1.1step 2.1

The completed space is compact. Fix $R > 1 / ε$ . The set $K : = {(x, y) \in X_{0} : ∣ x ∣ \leq R}$ is the intersection of the closed set $X_{0}$ with the closed cylinder ${∣ x ∣ \leq R}$ in $C^{2}$ , hence closed; on it $∣ y ∣^{2} = ∣ P (x) ∣ \leq \prod_{j} (R + ∣ a_{j} ∣)$ , so it is bounded in $C^{2}$ , hence compact [F7]. The image of the closed disc $∣ t ∣ \leq 1 / R$ under each of the two charts at infinity is compact, being a continuous image of a compact set [F7], and it contains the corresponding point $\infty_{\pm}$ ; by step 2.1 every point of $X$ with $∣ x ∣ > R$ lies in one of these two images. Therefore $X = K \cup (image of the + -chart) \cup (image of the - -chart)$ is a finite union of compact subsets, hence compact.

4.1F1F8step 1.1step 2.1step 2.2

The completed space is connected. By step 2.2 the set $Z$ is connected and contained in $X_{0}$ . Every point of $X_{0} ∖ Z$ , namely each $(a_{j}, 0)$ , is a limit point of $Z$ : in the chart $y \mapsto (ψ_{j} (y), y)$ of step 1.1 the points with $0 < ∣ y ∣ < δ$ have $x = ψ_{j} (y) = a_{j} + y^{2} w_{j} (y) \neq a_{j}$ , so they lie in $Z$ , and they tend to $(a_{j}, 0)$ as $y \to 0$ . Hence $X_{0}$ lies between the connected set $Z$ and its closure, so $X_{0}$ is connected [F8]. Likewise each $\infty_{\pm}$ is a limit point of $X_{0}$ : the points of its chart with $0 < ∣ t ∣ < δ$ belong to $X_{0}$ by step 2.1 and tend to $\infty_{\pm}$ as $t \to 0$ ; hence $X$ lies between the connected set $X_{0}$ and its closure, so $X$ is connected [F8]. By step 3.1, $X$ is a Riemann surface.

5.1F10step 1.1step 2.1step 3.1step 4.1

The ramification points of $π$ are the six branch points. At a point $(x_{0}, y_{0}) \in X_{0}$ with $y_{0} \neq 0$ the chart of step 1.1 has local parameter $x$ and the chart expression of $π$ towards $ϕ_{0}$ is $z \mapsto z$ , so the ramification index is $1$ there, by the description of the index as an order [F10]. Over each $a_{j}$ the only point of $X$ is $(a_{j}, 0)$ , because $y^{2} = P (a_{j}) = 0$ forces $y = 0$ ; in the chart with local parameter $y$ and the centred target chart at $a_{j}$ the expression of $π$ is $y \mapsto y^{2} w_{j} (y)$ with $w_{j} (0) \neq 0$ (step 1.1), whose order at $0$ is $2$ , so $e_{(a_{j}, 0)} (π) = 2$ [F10]. At $\infty_{\pm}$ the chart expression towards $ϕ_{\infty}$ is $t \mapsto t$ (step 2.1), so the index is $1$ and $\infty$ is not a branch value. Hence the branch values of $π$ are exactly the six distinct numbers $a_{1}, \dots, a_{6}$ , each with exactly one preimage, of index $2$ , and every other value has all its preimages of index $1$ .

6.1F7F11step 3.1step 3.2step 4.1step 5.1

$π$ is proper of degree two. By steps 3.1 and 4.1 the space $X$ is a Riemann surface and $π : X \to \hat{C}$ is nonconstant holomorphic. For compact $K \subseteq \hat{C}$ the preimage $π^{- 1} (K)$ is closed in $X$ , because $π$ is continuous and $K$ is closed in the Hausdorff space $\hat{C}$ ; being a closed subset of the compact space $X$ (step 3.2), it is compact [F7]. So $π$ is proper, and the degree theorem [F11] applies. For $b \in C ∖ {a_{1}, \dots, a_{6}}$ the fibre is $π^{- 1} (b) = {(b, \sqrt{P (b)}), (b, - \sqrt{P (b)})}$ , two distinct points, each of index $1$ by step 5.1, so $d = \deg π = 2$ .

7.1F5F6F7F12F13step 3.2step 4.1step 5.1step 6.1

The genus is two. By steps 3.2, 4.1 and 6.1 the map $π$ is a nonconstant holomorphic map of degree $2$ between compact connected Riemann surfaces, so Riemann-Hurwitz [F12] gives $2 g (X) - 2 = 2 (2 g (\hat{C}) - 2) + \sum_{x \in X} (e_{x} (π) - 1) .$ By step 5.1 the ramification points are exactly the six points $(a_{j}, 0)$ , each with index $2$ , while all other points, namely the affine points with $y \neq 0$ and the two points $\infty_{\pm}$ , have index $1$ ; hence the sum equals $6$ . The sphere $\hat{C}$ is compact, being homeomorphic to the closed bounded subset $S^{2}$ of $R^{3}$ [F5, F7], and connected [F6]; and $g (\hat{C}) = 0$ , because $\hat{C} ≅ S^{2} = #_{0} T^{2}$ and the genus is the unique handle number [F13]. Therefore $2 g (X) - 2 = 2 (0 - 2) + 6 = 2$ , that is $g (X) = 2$ .

8.1A1F14F17step 7.1

Hyperbolic type and the uniformization. By step 7.1 the surface $X$ is a compact Riemann surface of genus $2$ , so the compact-genus corollary [F14], whose choice hypothesis is covered by [A1], gives that $X$ has hyperbolic universal-covering type: its holomorphic universal cover $p : \tilde{X} \to X$ satisfies $\tilde{X} ≅ D$ [F14], and fixing a biholomorphism $ψ : \tilde{X} \to D$ gives a uniformization $(p, ψ)$ [F14]. By [F17] the conjugate $γ_{h} : = ψ \circ h \circ ψ^{- 1}$ is an automorphism of $D$ for every $h \in Deck (p)$ , and $Γ : = ψ Deck (p) ψ^{- 1} = {γ_{h} : h \in Deck (p)}$ is a subgroup of $Aut (D)$ , isomorphic to $Deck (p)$ .

9.1F16step 8.1

The transported covering, its deck group, and freeness. Define $Ψ : = p \circ ψ^{- 1} : D \to X$ . A homeomorphism of the total space carries evenly covered neighbourhoods to evenly covered neighbourhoods, so $Ψ$ is a covering map, with the same evenly covered sets as $p$ [F16]. A homeomorphism $h$ of $D$ satisfies $Ψ \circ h = Ψ$ exactly when $p \circ ψ^{- 1} h ψ = p$ , that is exactly when $ψ^{- 1} h ψ \in Deck (p)$ , that is exactly when $h \in Γ$ ; hence $Deck (Ψ) = Γ$ . Since $D$ is connected, deck transformations of $Ψ$ agreeing at one point are equal, so $Deck (Ψ) = Γ$ acts freely on $D$ [F16].

10.1F15F16step 9.1

The action of $Γ$ is properly discontinuous. Let $K \subseteq D$ be compact. Use the family of all evenly covered coordinate-disc neighbourhoods $U$ and smaller open neighbourhoods $W$ whose compact closures lie in $U$ . Finitely many $W_{i}$ cover $Ψ (K)$ . For each $i$ , the set $K \cap Ψ^{- 1} ({\overline{W}}_{i})$ is closed in $K$ , hence compact; the sheets over $U_{i}$ cover it, so only finitely many sheets meet it. Denote these by $S_{i}$ . If $γ K \cap K \neq \emptyset$ , write $γ x = y$ with $x, y \in K$ and choose $i$ with $Ψ (x) = Ψ (y) \in W_{i}$ . The sheets $V, V^{'} \in S_{i}$ containing $x, y$ are over the same $U_{i}$ , and $γ$ maps $V$ onto $V^{'}$ because it is a deck transformation. Two deck transformations mapping $V$ onto $V^{'}$ agree at the unique point of $V$ above any fixed base point, hence agree everywhere by [F16]. Therefore at most $\sum_{i} ∣ S_{i} ∣^{2}$ elements of $Γ$ move $K$ to meet itself, so the action is properly discontinuous [F15].

10.2F16F21step 9.1

$Γ$ is discrete. Fix $z_{0} \in D$ . For each $γ \in Γ$ choose an evenly covered neighbourhood of $Ψ (z_{0})$ and its sheet $V$ containing $γ (z_{0})$ . The compact-open set $S ({z_{0}}, V)$ is a neighbourhood of $γ$ . If $γ^{'} \in Γ$ also lies in it, then $γ^{'} (z_{0})$ and $γ (z_{0})$ lie in the same sheet and fibre, so injectivity on the sheet makes these values equal. Deck rigidity [F16] gives $γ^{'} = γ$ . Every element of $Γ$ is therefore isolated in the compact-open topology.

10.3F19F20step 9.1algebra

$Γ$ is torsion-free. Let $γ \in Γ$ with $γ^{m} = i d$ for some $m \geq 1$ ; we show $γ = i d$ . By [F19] there are $a \in D$ and $θ \in R$ with $γ = e^{i θ} φ_{a}$ , where $φ_{a} (z) = (a - z) / (1 - \overline{a} z)$ ; written as a quotient of linear polynomials this exhibits $γ$ as a Mobius transformation [F20]. Assume $γ \neq i d$ ; the classification [F20] gives two alternatives. In the parabolic alternative $γ$ is conjugate to $z \mapsto z + 1$ , so $γ^{m}$ is conjugate to $z \mapsto z + m \neq i d$ , contradicting $γ^{m} = i d$ . In the other alternative $γ$ has two fixed points and is conjugate to $z \mapsto λ z$ with $λ \in C^{\times} ∖ {1}$ ; then $γ^{m} = i d$ forces $λ^{m} = 1$ , so $∣ λ ∣ = 1$ , $λ \neq 1$ , and the invariant $τ (γ) = tr (A)^{2} / \det A$ of any representing matrix $A$ equals $λ + 2 + λ^{- 1} = 2 + 2 Re λ$ , which lies in $[0, 4)$ [F20]. The matrix $A = (\begin{array}{cc} - e^{i θ} & e^{i θ} a \\ - \overline{a} & 1 \end{array})$ represents $γ$ , and $tr A = 1 - e^{i θ}$ , $\det A = - e^{i θ} (1 - r^{2})$ with $r : = ∣ a ∣ < 1$ , so $τ (γ) = \frac{(1 - e^{i θ})^{2}}{- e^{i θ} (1 - r^{2})} = \frac{4 \sin^{2} (θ / 2)}{1 - r^{2}} .$ Hence $4 \sin^{2} (θ / 2) < 4 (1 - r^{2})$ , that is $r^{2} < \cos^{2} (θ / 2)$ ; writing $c : = \cos (θ / 2)$ we have $r < ∣ c ∣$ . If $a = 0$ then $γ (z) = - e^{i θ} z$ fixes $0 \in D$ , contradicting the freeness of the action of $Γ$ (step 9.1); so $a \neq 0$ and $r > 0$ . The fixed points of $γ$ solve $γ (z) = z$ , that is $\overline{a} z^{2} - (1 + e^{i θ}) z + e^{i θ} a = 0$ ; substituting $z = e^{i θ / 2} w$ and dividing by $e^{i θ}$ turns this into $\overline{a} w^{2} - 2 c w + a = 0$ , and multiplying by $a$ gives $r^{2} w^{2} - 2 a c w + a^{2} = 0$ , whose roots are $w = a (c \pm s) / r^{2}$ with $s : = (c^{2} - r^{2})^{1 / 2} > 0$ real. Hence $z_{\pm} = e^{i θ / 2} a (c \pm s) / r^{2}$ are the two fixed points of $γ$ , and $∣ z_{\pm} ∣ = ∣ c \pm s ∣ / r$ . Since $c^{2} - s^{2} = r^{2} > 0$ , one has $s < ∣ c ∣$ and the two numbers $c \pm s$ have the same sign. Thus $\min {∣ z_{+} ∣, ∣ z_{-} ∣} = (∣ c ∣ - s) / r$ . Moreover $∣ c ∣^{2} - s^{2} = r^{2}$ , so $(∣ c ∣ - s) / r = r / (∣ c ∣ + s) < 1$ . Hence one of the fixed points lies in $D$ , contradicting freeness (step 9.1).

10.4F18F22step 3.2step 9.1

$D / Γ$ is homeomorphic to $X$ and compact. Let $q : D \to D / Γ$ be the quotient map of the action of $Γ$ and give $D / Γ$ the quotient topology [F22]. For $γ_{h} \in Γ$ one has $Ψ (γ_{h} z) = p (ψ^{- 1} ψ h ψ^{- 1} z) = p (h ψ^{- 1} z) = p (ψ^{- 1} z) = Ψ (z)$ , so $Ψ$ is $Γ$ -invariant and induces a map $Φ : D / Γ \to X$ with $Φ \circ q = Ψ$ ; by the characteristic property of the quotient topology $Φ$ is continuous [F22]. It is surjective because $Ψ$ is. It is injective: if $Ψ (z) = Ψ (z^{'})$ , then $ψ^{- 1} z$ and $ψ^{- 1} z^{'}$ lie in one fibre of $p$ , and the deck group of the universal cover acts transitively on each fibre [F18], so $ψ^{- 1} z^{'} = h ψ^{- 1} z$ for some $h \in Deck (p)$ and hence $z^{'} = γ_{h} z$ , that is $q (z^{'}) = q (z)$ . The map $Ψ$ is open: if $O \subseteq D$ is open and $w = Ψ (u) \in Ψ (O)$ , choose an evenly covered $U ∋ w$ with sheet $V ∋ u$ ; then $V \cap O$ is open and $Ψ (V \cap O)$ is open in $U$ , hence in $X$ , and contains $w$ . Consequently for every open $O \subseteq D / Γ$ the set $Φ (O) = Ψ (q^{- 1} (O))$ is open in $X$ [F22], since $q$ is surjective; so the continuous bijection $Φ$ is a homeomorphism [F22]. Therefore $D / Γ ≅ X$ is compact by step 3.2, that is, $Γ$ is cocompact.

11.1step 3.2step 4.1step 5.1step 6.1step 7.1step 8.1step 9.1step 10.1step 10.2step 10.3step 10.4∎

Conclusion. Steps 3.1 to 4.1 exhibit $X$ as a compact connected Riemann surface (steps 3.2 and 4.1). Step 6.1 shows that the projection $π : X \to \hat{C}$ is a proper holomorphic map of degree $2$ ; step 5.1 identifies its branch values as the six numbers $a_{1}, \dots, a_{6}$ , each with a single point of index $2$ and with $\infty$ unramified; and step 7.1 computes $g (X) = 2$ . By step 8.1 the surface $X$ has hyperbolic universal-covering type, with uniformization $Ψ : D \to X$ whose deck group is $Γ \leq Aut (D)$ (step 9.1). The group $Γ$ acts freely (step 9.1) and properly discontinuously (step 10.1), so it is Fuchsian in the sense of the Example; it is torsion-free (step 10.3) and discrete for the compact-open topology (step 10.2); and $D / Γ ≅ X$ is compact (step 10.4), so the quotient is cocompact. This proves all the assertions of the Example.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Compactness and Liouville distinguish the three models

Example

Let $\hat{C}$ be the Riemann sphere, $C$ the complex plane and $D = {z : ∣ z ∣ < 1}$ the unit disc, each with its usual topology and complex structure. The three models are pairwise non-biholomorphic, and the two available reasons are independent of one another:

$\hat{C}$ is compact while $C$ and $D$ are not, so no homeomorphism, and hence no biholomorphism, can join the sphere to either of the other two.
A biholomorphism $C \to D$ would be a bounded entire function that is not constant, which Liouville's theorem forbids.

The second obstruction is genuinely complex-analytic: $C$ and $D$ are homeomorphic (both are homeomorphic to $R^{2}$ ), so topological type alone does not determine complex structure.

Facts & Assumptions

Given: The Riemann sphere, the complex plane and the unit disc with their usual topologies and complex structures. Here a biholomorphism between Riemann surfaces means a bijective holomorphic map with holomorphic inverse, with holomorphicity understood chartwise as in [F7].

[F1]

The sphere, the plane and the disc are simply connected Riemann surfaces, and no two of them are biholomorphic (The sphere, plane and disc are pairwise biholomorphically distinct).

[F2]

Every bounded entire function is constant: if $f : C \to C$ is holomorphic and $∣ f (z) ∣ \leq M$ for all $z$ and some real $M \geq 0$ , then $f$ is constant (Liouville's theorem: every bounded entire function is constant).

[F3]

A space is compact when every open cover of it has a finite subcover (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F4]

For $n \geq 1$ the Euclidean closed balls and spheres in $R^{n}$ are compact (For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[F5]

Continuous images of compact sets are compact: for a continuous $f : X \to Y$ and compact $K \subseteq X$ the image $f [K]$ is a compact subset of $Y$ (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F6]

Stereographic projection $Σ : \hat{C} \to S^{2}$ is a homeomorphism onto the unit sphere (Stereographic projection identifies the Riemann sphere with the unit two-sphere).

[F7]

A holomorphic map of Riemann surfaces is continuous (Holomorphic maps and meromorphic functions on Riemann surfaces).

[F8]

For complex domains, a map is biholomorphic when it is bijective, holomorphic, and has holomorphic inverse (Biholomorphic maps between complex domains).

Proof technique: direct: exhibit the explicit open covers that fail to have finite subcovers, and the explicit bounded nonconstant entire function that Liouville's theorem excludes.

Verification

1.1F4F5F6

The sphere $\hat{C}$ is compact: $Σ$ is a homeomorphism onto $S^{2}$ [F6], the sphere $S^{2} \subseteq R^{3}$ is compact [F4], and $Σ^{- 1}$ is continuous, so $\hat{C} = Σ^{- 1} (S^{2})$ is a continuous image of a compact set [F5].

1.2F3algebra

Neither $C$ nor $D$ is compact. The open discs $D (0, n)$ , $n \geq 1$ , cover $C$ ; any finitely many of them are contained in $D (0, N)$ for $N$ the largest index occurring, which omits every point of modulus greater than $N$ , so no finite subfamily covers $C$ . Likewise the open discs $D (0, 1 - 1 / n)$ , $n \geq 2$ , cover $D$ ; any finitely many are contained in $D (0, 1 - 1 / N)$ for $N$ the largest index occurring, which omits the points of modulus between $1 - 1 / N$ and $1$ , so no finite subfamily covers $D$ . By the definition of compactness neither space is compact.

1.3F2F8algebra

The plane is not biholomorphic to the disc: if $f : C \to D$ were a biholomorphism, then by [F8] $f$ is holomorphic and bijective, and regarding it as a map into $C$ it is entire with $∣ f (z) ∣ < 1$ for every $z$ ; by [F2] such an $f$ must be constant, and a constant map is not injective, hence not bijective, a contradiction. So no biholomorphism $C \to D$ exists.

2.1F3F5F7givenstep 1.1step 1.2

Suppose there were a biholomorphism $f : \hat{C} \to C$ ; by the given meaning of biholomorphism and [F7], both it and its inverse are continuous, so it is a homeomorphism and is surjective onto $C$ . Since $\hat{C}$ is compact by step 1.1, its continuous image $C$ would be compact [F5], contradicting step 1.2. The same argument with $D$ in place of $C$ excludes a biholomorphism $\hat{C} \to D$ . So compactness separates the sphere from the plane and the disc.

3.1F1step 1.1step 1.2step 2.1step 1.3algebra∎

The obstruction in step 1.3 is not topological. The map $Φ (z) : = z / (1 + ∣ z ∣)$ is a continuous bijection of $C$ onto $D$ , because $∣ Φ (z) ∣ = ∣ z ∣ / (1 + ∣ z ∣) < 1$ with equality approached but never attained, and its inverse is $Φ^{- 1} (w) = w / (1 - ∣ w ∣)$ , also continuous; so $C$ and $D$ are homeomorphic. Nevertheless step 1.3 shows they are not biholomorphic, while [F1] independently records the pairwise non-bihomorphism of all three models. Hence the two distinctions exhibited above — compactness for the sphere, Liouville for the plane versus the disc — are the classical witnesses for the inequivalence of the three simply connected models. Every cover and every map used is given by an explicit formula, so no choice principle is used.

Hyperbolic Riemann Surfaces and Uniformization: Examples and Counterexamples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Hyperbolic distances and geodesics in disc and half-plane

Example

Facts & Assumptions

Verification

Annulus and punctured disc have hyperbolic universal covers

Example

Facts & Assumptions

Verification

A complex torus has a lattice of parabolic deck translations

Example

Facts & Assumptions

Verification

A genus-two compact surface gives a cocompact Fuchsian group

Example

Facts & Assumptions

Verification

Compactness and Liouville distinguish the three models

Example

Facts & Assumptions

Verification

Sources