Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Overview of Riemann Surfaces
Norio Kosaka
Department of Economics, Mathematics and Statistics
Birkbeck, University of London

22 September 2023

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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Objective of Thesis

Properly define Riemann Surfaces to;▶ (i) State the Uniformisation Theorem to classify the surfaces▶ (ii) Determine / Study their automorphism groups and theirimportance▶ (iii) Study the geometric representation of critical points

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Preliminary: Manifolds
Start with reviewing a couple of concepts.
Recall: Atlas
A set of charts A := {(Ui , ϕi ) | ϕi : Ui → D ⊂SC} is called anatlas on a surface R if the following hold: (i) (Ui ,ϕi )∈A Ui = R,and (ii) transition among charts in the atlas is analytic.

When multiple atlases have analytic transition functions, then they
are called compatible.
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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Formally, Riemann Surfaces are ...
The compatibility of atlases forms an equivalence relation (Exercise4F of [JS87]) that forms the complex structure on R as follows;
Def: Conformal Structure
A conformal structure on R is an atlas A on R which is maximal:if (ψ, V ) is a chart on R such that, for any (ϕ, U) ∈ A, if it iscompatible with (ψ, V ), then (ψ, V ) ∈ A.
We now have all the concepts we need to define Riemann surfaces.
Def: Riemann Surfaces
A Riemann surface is a pair (R, A), where A is a conformalstructure on R.

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’EnfantsReferences

Example: Riemann Sphere

A surface S = Σ := C ∪ {∞} ishomeomorphic to the sphere S 2 withstereographic projection.

Suppose two charts: Identity map (ϕ1 : z 7→ z on C) andReciprocal map (ϕ2 : z 7→ 1/z with ϕ2 (∞) = 0 on Σ \ {0}).
We have (ϕ2 ◦ ϕ−11 )(z) = 1/z which is analytic on
ϕ1 (U1 ∩ U2 ) = C \ {0} and similarly for (ϕ1 ◦ ϕ−12 ). Thus, this is a
Riemann surface, called Riemann sphere.
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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

To the next...

Now, we will take a look at cases where Riemann surfaces are
simply connected and the classification results of those surfaces.
1. Analytic continuation
2. Covering space

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References(1) Analytic continuation and Example
Analytic continuation extends the representation of a function inone region of the complex plane into another region.
Ex: For z = re iθ , let f (z) = log (z) = ln r + i(θ + 2nπ) with n ∈ Z.
2D3D𝑦𝜃 = 2𝑛𝜋
Escalator𝑥𝜃 = 2(𝑛 + 1)𝜋Escalator

Continuation on a simply connected region E is path-independent(w/h common start/end points) by Monodromy theorem.
So we have a single-valued analytic function on E over different
function elements.
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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References(2) Covering of a Riemann surface: Example
Let f (z) = z n . We have f (0) = 0 with multiplicity n, and asf ′ (z ̸= 0) ̸= 0, there are no other branch-points in C. Similarly for∞ ∈ Σ, it has the multiplicity of n and thus a branch-point.
Hence, n-sheets come together at branch-points of 0 and ∞.

Figure: When n = 2.
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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References(2) Covering of a Riemann surface: Theorem

Indeed, for connected surfaces, the covering spaces turn out to be
simply connected as promised.
Thm: Regular covering space; ((10.19) of [Arm13])
Every connected surface S has a covering surface (S̃, p) where p isa covering map such that S̃ is simply connected.

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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Classify simple regions: Riemann Mapping Thm

Conformal equivalence
A conformal equivalence is an analytic bijection of Riemannsurfaces f : R → S with any analytic inverse f −1 : S → R.
Ex: Möbius transformations.
Riemann Mapping Theorem classifies all simply connected open
subsets of C up to conformal equivalence.
Riemann Mapping Theorem (Corollary 16.15 of [Wil20])
If S is a simply connected open subset of C, then either S = C orelse S ≃ D (Unit disc).

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Statement / Insight of Uniformisation theorem
Uniformisation Theorem (Theorem 4.17.2 of [JS87])
Every simply connected Riemann surface is conformally equivalent
to one of Σ = C ∪ {∞} (Riemann sphere), C (Complex plane), or
D (Disc).
Remark
The three Riemann surfaces are not conformally equivalent toeach other;▶ Σ is compact, so not even homeomorphic to the other two,
▶ D is bounded as a disc and by Liouville’s theorem any analytic
map C → D is constant. Thus, not conformally equivalent as
not bijection.
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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Uniformisation and Coverings

Relationship between a Riemann surface and the covering space isindeed profound.
Corollary 16.5 of [Wil20] or Theorem 4.19.5 of [JS87].
Every Riemann surface R is conformally equivalent to a quotientR ≃ R̃/G where R̃ is one of Σ, C or D, and G is a (properlydiscontinuous) group of conformal equivalences (automorphisms)of R̃.
Thus, it is important to determine the automorphism groups of theRiemann surfaces to study Uniformisation theorem.

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Automorphism groups
Following shows the classification of automorphism groups.
Thm 4.17.3 of [JS87]
1. Aut(Σ) = PSL(2, C) (Möbius transformations),
2. Aut(C) = {z 7→ az + b | a, b ∈ C, a ̸= 0} (Linear transf.),
3. Aut(H) = PSL(2, R) where H is the upper half-plane.
RemarkD (Disc) and H are conformally equivalent via a Möbiustransformation T ∈ PSL(2, C) : H → C such that z 7→ z−iz+i(Example (1) of Sec4.17 of [JS87]).

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Automorphism groups

In fact, we can find a stronger result helping us to study ofclassifications of Riemann surfaces.
Mutual Exclusivity: Prop. 16.10 of [Wil20]
A Riemann surface R is uniformised by at most one of Σ, C and D.
Now let us focus on the upper half-plane and the associatedactions, i.e., the automorphisms group PSL(2, R), namely
Fuchsian groups.

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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

What’s PSL(2, R)?
Intuitively, we can think of Möbius transformations in PSL(2, C)with the Euclidean distance.
Möbius transformations
az+b
It is a complex rational function of the form; f (z) = cz+d
forz, a, b, c, d ∈ C such that ad − bc ̸= 0.

RemarkMöbius transformations can take various forms;▶ Translation: f (z) = b + z (a = 1, c = 0, d = 1),▶ Rotation: f (z) = az (b = 0, c = 0, d = 1),▶ Inversion: f (z) = 1/z (a = 0, b = 1, c = 1, d = 0),
Caution: PSL(2, R) acts with the distance defined on a Hyperbolic
space model (H).
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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Preliminary to Fuchsian groups
Def: Properly discontinuous actions
Let a group G act by homomorphism on a space X . The action issaid to be properly discontinuous if, for every compact K ⊆ X ,the set {g ∈ G | g (K ) ∩ K ̸= ∅} is finite.
For instance, if Ω = Ω(ω1 , ω2 ) is a lattice in C then the action ofΩ on C by translation is properly discontinuous. By tessellation,
we can generate the torus (a quotient space of C/Ω)
𝑤

𝑤 +𝑤
𝑤 =𝑤 +𝑤
0=𝑤

0𝑤
0=𝑤 =𝑤=𝑤 +𝑤
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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Fuchsian groups

Fuchsian groups
A subgroup of PSL(2, R) that acts properly discontinuously on H
is called a Fuchsian group.
Examples of Fuchsian groups▶ Integer translations T (z) = {z + n | n ∈ N} form a Fuchsiangroup.
▶ More generally, as SL(2, Z) ⊂ SL(2, R), PSL(2, Z) (Modulargroup) is a subgroup of PSL(2, R) and hence is a Fuchsiangroup.

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Fuchsian groups and Uniformisation Theorem
Similar to the quotient space
Q of the torus (C/Ω), we can define aquotient space H/Γ with: H → H/Γ s.t. z 7→ [z]Γ and [z]Γ isthe Γ-orbits.
The following result shows that the quotient-spaces of Fuchsiangroups are Riemann surfaces.
Thm 5.9.1 of [JS87]
H/Γ is a connected Riemann surface withholomorphic.

Q: H → H/Γ being

In summary, the problem of classifying the Riemann surfaces
uniformised by D involves first classifying Fuchsian groups, andthen understanding their properly discontinuous actions.
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Norio Kosaka

Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’EnfantsReferences

Outline▶ Objective of Thesis▶ Riemann surfaces▶ Uniformisation theorem▶ Fuchsian groups▶ Dessins d’Enfants

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Introduction

Now, we turn our attention to interpreting the critical points of a
Riemann surface to develop more intuition.
Recall: Critical points
A critical point is a point in the domain of the function where thefunction is either not differentiable or the derivative is equal tozero.

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Belyi maps
In the following, ”a compact Riemann surface S is defined over Q̄”(Algebraic closure of Q) means the corresponding algebraic curveis defined over Q̄.
Belyi’s theorem: Thm 5.1 of [Pos+14] and Sec.2 of [Pér18]
A compact Riemann surface S can be defined over Q̄ if and only ifthere exists a covering β : S → Σ unramified outside of {0, 1, ∞}.
Belyi map
A rational function β : S → Σ which has at most three criticalvalues {0, 1, ∞} is called a Belyi map where S is a compact
Riemann surface. (S, β) is called a Belyi pair.
Example: β(z) = f (z) = z n from the slide.11.
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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Definition of Dessins
In the following, imagining a bipartite graph on a surface would beuseful.
For a Belyi map β : S → Σ, we define the preimages;β −1 ({0})β −1 ({1})β −1 ({[0, 1]})

{black vertices}{white vertices}{edges}The bipartite graph ∆β = (V , E ) with the coloured vertices andthe edges (called Hypermap) is called a Dessin d’Enfant.
Intuitively, we embed the graph on the surface S in 3-dimension tovisualise.
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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

Example: Computing a Dessin
Task: Figure out the underlying Belyi map given the following
figure with critical points and one unknown (Vertex a). Note thatthe black vertices correspond to the fibre f −1 ({0}).

10𝑎∞Observe that;▶ Vertex 1 has degree 3 and Vertex a has degree 1,▶ Locations of Vertices 0, ∞ imply the pole at x = 0 or x = ∞.
We can infer the form of the underlying Belyi map f as;
f (x) = K

(x − 1)3 (x − a)for a constantK .
x
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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groupsDessins d’Enfants

References

Example: Computing a Dessin
Define the white vertices of f on this graph by observing;▶ f − 1 forms (a) roots on f −1 ({1}),▶ As bipartite, two white vertices cannot be on the same edge!
Thus, the placements of white vertices are as follows;

1𝑎And this implies the double roots, so we obtain;f (x) − 1 = K(x 2 + bx + c)2
for K , b, c ∈ R.
x(1)
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Obj. of Thesis

Riemann surfaces

Uniformisation theoremFuchsian groups

Dessins d’Enfants

References

Example: Computing a Dessin
Substituting the definition of f (x) into (1) and working out theequation, we get a = 9, b = −6, c = −3, and K = −1/64. Thus,(x 2 − 6x − 3)2(x − 1)3 (x − 9)
, f (x) − 1 = −64x64x√
The white vertices are the roots of x 2 − 6x − 3, that is 3 ± 2 3.
f (x) = −13 − 2√3

3 + 2√3

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Obj. of Thesis

Riemann surfaces

Uniformisation theoremFuchsian groups

Dessins d’Enfants

References

A result: Grothendieck correspondence

Let us conclude the talk by introducing one result.
The following result states our intuition from the previous example;
Grothendieck correspondence (Thm 6.7 of [Pos+14])
There is a one-to-one correspondence between Dessins and Belyi
pairs.

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Overview of Riemann Surfaces

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Obj. of Thesis

Riemann surfaces

Uniformisation theorem

Fuchsian groups

Dessins d’Enfants

References

References[JS87]

Gareth A Jones and David Singerman. Complex
functions: an algebraic and geometric viewpoint.
Cambridge university press, 1987.

[Arm13]

Mark Anthony Armstrong. Basic topology. Springer
Science & Business Media, 2013.

[Pos+14]

DPA van der Post et al. “Riemann surfaces and dessinsd’enfants”. B.S. thesis. 2014.

[Pér18]

Javier Alcaide Pérez. “Riemann surfaces and dessind’enfants”. PhD thesis. 2018.

[Wil20]Henry Wilton. Lecture notes in Riemann Surfaces.
2020.
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