Examensarbete

On Poincar´esUniformization Theorem Gabriel Bartolini

LITH-MAT-EX–2006/14

On Poincar´esUniformization Theorem

Applied Mathematics, Link¨opingsUniversitet

Gabriel Bartolini

LITH-MAT-EX–2006/14

Examensarbete: 20 p

Level: D

Supervisor: Milagros Izquierdo Barrios, Applied Mathematics, Link¨opingsUniversitet Examiner: Milagros Izquierdo Barrios, Applied Mathematics, Link¨opingsUniversitet Link¨oping: December 2006

Avdelning, Institution Datum Division, Department Date Matematiska Institutionen December 2006 581 83 LINKOPING¨ SWEDEN

Spr˚ak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling x Engelska/English x Examensarbete LITH-MAT-EX–2006/14 C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering Ovrig¨ rapport

URL f¨orelektronisk version

Titel On Poincar´esUniformization Theorem Title

F¨orfattare Gabriel Bartolini Author

Sammanfattning Abstract A compact Riemann surface can be realized as a quotient space U/Γ, where U is the sphere Σ, the euclidian plane C or the hyperbolic plane H and Γ is a discrete group of automorphisms. This induces a covering p : U → U/Γ. For each Γ acting on H we have a polygon P such that H is tesselated by P under the actions of the elements of Γ. On the other hand if P is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group Γ generated by the side pairing is discrete and P tesselates H under Γ.

Nyckelord Keyword Hyperbolic plane, automorphism, Fuchsian group, Riemann surface, cov- ering, branched covering, orbifold, uniformization, fundamental domain, Poincar´estheorem vi Abstract

A compact Riemann surface can be realized as a quotient space U/Γ, where U is the sphere Σ, the euclidian plane C or the hyperbolic plane H and Γ is a discrete group of automorphisms. This induces a covering p : U → U/Γ. For each Γ acting on H we have a polygon P such that H is tesselated by P under the actions of the elements of Γ. On the other hand if P is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group Γ generated by the side pairing is discrete and P tesselates H under Γ.

Bartolini, 2006. vii viii Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Chapter outline ...... 2

2 General Setting 3 2.1 The Extended Complex Plane ...... 3 2.1.1 The Riemann Sphere ...... 3 2.1.2 M¨obiusTransformations ...... 4 2.2 Surfaces ...... 7 2.2.1 Riemann Surfaces ...... 7 2.2.2 Automorphisms ...... 9 2.3 The Hyperbolic Plane ...... 10 2.4 Fuchsian groups ...... 14

3 Covering Maps 17 3.1 Fundamental Groups ...... 17 3.2 Group Actions on Surfaces ...... 20 3.3 Universal Coverings ...... 22

4 Riemann Surfaces as Orbifolds 25 4.1 2-Orbifolds ...... 25 4.2 Branched Coverings ...... 26 4.2.1 Universal Branched Coverings ...... 27

5 Poincar´esTheorem 31 5.1 Fundamental Domains ...... 31 5.2 The Quotient Space H/Γ...... 32 5.3 Poincar´esTheorem ...... 34

Bartolini, 2006. ix x Contents List of Figures

2.1 Stereographic projection ...... 4 2.2 Hyperbolic lines in H and D...... 12 2.3 Hyperbolic triangle ...... 13 2.4 Hyperbolic and parabolic elements...... 15 2.5 Triangle with reflections ...... 16

3.1 A torus ...... 19 3.2 A double torus ...... 20 3.3 A torus as a quotient space ...... 20

4.1 The pillow case ...... 28

5.1 Fundamental domain for a triangle group ...... 32 5.2 Hyperbolic triangle ...... 35 5.3 Hyperbolic triangle with reflections ...... 36 5.4 Path lifting with triangles ...... 38 5.5 Fundamental polygon ...... 40 5.6 The poincar´edisc tesselated by the modular group [4] . . . . . 41 5.7 The poincar´edisc tesselated by the (2, 3, 7) triangle group [4] . 42

Bartolini, 2006. xi xii List of Figures Chapter 1

Introduction

1.1 Background

The study of Riemann surfaces begun in the 19th century when Riemann introduced them in his doctoral dissertation Foundations for a general theory of functions of a complex variable. An important perspective in the study of Riemann surfaces is the con- cept of uniformization, which was developed by Poincar´e,Klein and others. The theory states that every closed orientable Riemann surface admits a Riemann metric of constant curvature. The main geometries are the sphere, the Euclidian plane and the hyperbolic plane with curvature 1, 0 and −1 respectively. Poincar´ediscovered that groups of M¨obiustransformations preserved the structures of the non-Euclidian geometries. The Riemann surfaces can be realized as quotient spaces U/Γ where U is either the sphere, the Euclidian plane or the hyperbolic plane and Γ is a discrete group of structure preserving M¨obiustransformations. Poincar´esclassical theorem states that a hyperbolic polygon satisfying certain conditions is a fundamental domain for a Fuchsian group acting on the hyperbolic plane. Poincar´egeneralized the theorem to polyhedra in 3- dimensional hyperbolic space where the groups are Kleinian rather that Fuch- sian. The 3-dimensional manifolds are much more complicated than surfaces, but Thurston showed a connection between the 3-dimensional case and the 2-dimensional. There are still open questions in the 3-dimensional case. Here we will look at the 2-dimensional case. Several proofs of Poincar´es theorem have been published, but their validity has been questioned. We will limit ourselves to the case where the polygon is a triangle. The discrete groups associated with triangles, known as triangle groups, where introduced

Bartolini, 2006. 1 2 Chapter 1. Introduction

by Schwarz.

1.2 Chapter outline

Chapter 2 Here we introduce the concepts of the non-euclidian geometries, in particular the hyperbolic plane as well as the automorphism groups of them. We also construct Riemann surfaces and the functions defined on them.

Chapter 3 In our study of surfaces we need some basic topological tools. We look at coverings and homotopy and how homotopy classes of paths are related to coverings.

Chapter 4 In chapter 3 we only used the topological structures of surfaces. In order to include the complex structures of Riemann surfaces we need to introduce the concept of orbifolds and its relation to Riemann surfaces. We will also look at the concept of uniformization for Riemann surfaces.

Chapter 5 Here we construct the fundamental domains for Fuchsian groups and look at the quotient space H/Γ. Lastly we will look at Poincar´esclas- sical theorem for polygons and make a complete proof for hyperbolic triangles.

The theory of Riemann surfaces, M¨obiustransformations and Fuchsian is found in [1], [2] and [8]. The coverings are found in [3] and branched coverings and orbifolds are based on [3] and [6]. Fundamental domains are found in [1], [2] and the proof of Poincar´estheorem is based on the ideas used in [1], [5] and [7]. Chapter 2

General Setting

First we need to construct the non-euclidian geometries we need for our later work. We will also look at M¨obiustransformations wich will play an important role.

2.1 The Extended Complex Plane

We begin with the extended complex plane which will be our model for the spherical geometry. To better understand the extended complex plane and M¨obiustransformations we will start with constructing what is called The Riemann Sphere. We do this by stereographic projection.

2.1.1 The Riemann Sphere Consider the units phere in R3 given by

2 2 2 x1 + x2 + x3 = 1. (2.1)

We want to associate a point on the sphere with a point in the complex plane C. Think of C as the plane given by x3 = 0. Now take the line that passes through N = (0, 0, 1) and a point z ∈ C. This line intersects the sphere at a single pointz ˆ, which is the stereographic projection of z. Now if z = x + iy andz ˆ = (x1, x2, x3) then we have

2x 2y |z|2 − 1 x = , x = , x = (2.2) 1 |z|2 + 1 2 |z|2 + 1 3 |z|2 + 1 and x x x = 1 , y = 2 . (2.3) 1 − x3 1 − x3

Bartolini, 2006. 3 4 Chapter 2. General Setting

zˆ C

z

Figure 2.1: Stereographic projection

Now we look at a nice property of the Riemann sphere. Lines and circles in C are given by an equation of the form

a(x2 + y2) + bx + cy + d = 0. (2.4)

With the formulas 2.1 and 2.3 we can rewrite this as

bx1 + cx2 + (a − d)x3 + a + d = 0. (2.5)

This is the equation of a plane in R3 and the intersection of a plane and a sphere is a circle. This means that all lines and circles in C are projected to circles on the sphere. Note that the lines are projected to circles through N. With this in mind we let Σ = C ∪ {∞}, known as the extended comlex plane and we project ∞ on N. With stereographic projection and the Riemann sphere we will think of Σ as a sphere and lines as circles through ∞, or N. This will make some of the properties of M¨obiusTransformations easier to understand.

2.1.2 M¨obiusTransformations A M¨obiusTransformation on Σ is a function of the form az + b T (z) = , a, b, c, d ∈ C, ad − bc 6= 0 (2.6) cz + d where that last requirement is to make sure T is not constant. We also set T (∞) = a/c and T (−d/c) = ∞ if c 6= 0 else T (∞) = ∞. All M¨obiusTransformations can be written as a composition of (i) translations z 7→ z + c c ∈ C

(ii) rotations z 7→ eiφz 0 ≤ φ ≤ 2π 2.1. The Extended Complex Plane 5

(iii) magnifications z 7→ λz λ ∈ R+

1 (iv) inversion z 7→ z 2 iπ 2 Example 1 Let T (z) = . Let T0(z) = e z = −z then T (z) = . −z+i T0(z)+i 1 Similarly let T1(z) = z + i, T2(z) = z and T3(z) = 2z then we see that T (z) = T3(T2(T1(T0(z)))) = T3T2T1T0(z). It’s easy to see that all translations, rotations, magnifications and inver- sion map Σ bijectively to itself. Remenbering that lines are circles on Σ we also see that all those transform circles to circles. Example 2 Let L = {z = x + iy ∈ Σ|ax + by = c 6= 0}, that is a line not passing through origo, then 1 1 1 w = ⇔ z = x + iy = = z w u + iv au −bv a b c = ax + by = + ⇔ u2 + v2 − u + v = 0, u2 + v2 u2 + v2 c c where the last equation gives a circle. As we know from geometry three noncollinear points uniquely determines a circle. If they’re collinear they uniquely determines a line. We can map any circle on Σ to any other circle. To see this we choose three distinct points z1, z2, z3 ∈ Σ. If all points are finite, let (z − z )(z − z ) T (z) = 1 2 3 (2.7) (z − z3)(z2 − z1) which satisfies T (z1) = 0,T (z2) = 1,T (z3) = ∞. If any of the zi is ∞ then we get the same result by removing the factors containing that point. If w1, w2, w3 ∈ Σ and and S is a M¨obiustransformation such that S(w1) = −1 0,S(w2) = 1,S(w3) = ∞ we see that S T maps zi to wi for each i. −1 Assme that U also maps z1, z2, z3 to 0, 1, ∞ respectively then UT fixes −1 az+b −1 0, 1, ∞. Now let UT (z) = cz+d . Solving UT (zi) = zi for i = 1, 2, 3 gives a = d 6= 0, b = c = 0, that is UT −1 = Id and U = T . So T is unique. It follows that there’s is a unique transformation mapping zi to wi i = 1, 2, 3. Example 3 The M¨obiusTransformation mapping R∪{∞} to the unit circle with T (0) = −1,T (1) = −i, T (∞) = 1 is given by z − i T (z) = . (2.8) z + i We also note that the upper half plane is mapped to the interior of the circle. 6 Chapter 2. General Setting

We want to work with the M¨obiustransformations as a group. We do this by representing a transformation T with a matix A. The general¡ linear¢ a b group GL(2, C) is the group of invertible 2x2-matrices. Now if A = c d ∈ az+b GL(2, C), we let TA(z) = cz+d . We see that for k 6= 0, TkA = TA. So we can allways choose a, b, c, d ∈ C such that ad − bc = 1. This restriction gives us the special linear group SL(2, C). We also note that if A ∈ SL(2, C) then −A ∈ SL(2, C) and T−A = TA as seen before. Identifying A and −A gives us the special projective group PSL(2, C). Unless other noted, we will think of M¨obiustransformations as elements of PSL(2, C). Two elements g1, g2 of a group G are called conjugate if there exists −1 h ∈ G such that g2 = hg1h . Cunjugacy is an equivalence relation and the equivalence classes are called conjugacy classes. It will be useful to divide PSL(2, C) into conjugacy classes but first we need to look at a particular property of elements is PSL(2, C). A fixed point of an element T is a point −1 z0 such that T (z0) = z0. We see that S(z0) is a fixed point of STS , this means that conjugate elements of PSL(2, C) have the same number of fixed points. The conjugacy classes are distinguished by their fixed points. We have seen that only the identity has three (or more) fixed points. Now let T = (az + b)/(cz + d) ∈ PSL(2, C).If c 6= 0 then z ∈ C is a fixed point of T if and only if cz2 + (d − a)z − b = 0 which has two solutions unless (a + d)2 = 4. If c = 0 then ∞ is a fixed point of T and ad = 1 which gives T (z) = a2z + ab. There is a second fixed point if and only if a2 6= 1, that is (a + d)2 6= 4. Let ±A ∈ SL(2, C) be the pair of matrices associated with T . The trace tr(A) = a + d is invariant under conjugation and tr(−A) = −tr(A) so tr2(±A) = (a + d)2 depends only on the conjugacy class of T .

Theorem 4 Two M¨obiustransformations T1,T2 6= Id are conjugate if and 2 2 only if tr (T1) = tr (T2). With those results we divide PSL(2, C) in classes. Let T ∈ PSL(2, C), then T is

(i) parabolic if (a + d)2 = 4. T has one fixed point and is conjugate to z + 1

(ii) hyperbolic if (a + d)2 > 4. T has two fixed points and is conjugate to λz (|λ| 6= 1, λ ∈ R)

(iii) elliptic 0 ≤ (a + d)2 < 4. T has two fixed points and is conjugate to λz (λ = eiθ) 2.2. Surfaces 7

(iv) loxodromic if (a + d2 < 0) or (a + d)2 ∈/ R. T has two fixed points and is conjugate to λz (|λ|= 6 1, λ∈ / R)

2.2 Surfaces

Before we look at some additional properties of M¨obiustransformations, we have to look at surfaces and functions on surfaces. First we need some types of functions on C.

2.2.1 Riemann Surfaces

Let γ1, γ2 be two paths [0, 1] → C passing trough a point p ∈ C. Let θ be the angle between the tangets of γ1, γ2 at p. We say that γ1, γ2 intersect with angle θ at p. A function f is called conformal at p if f(γ1), f(γ2) intersects with angle θ at f(p). We call a function f : U → C, where U ⊂ C, analytic on U if f 0(z) exists for all z ∈ U. When working with Riemann surfaces we will encounter singularities. There are three types of singularities. Let f : U → C be analytic on U \{p} where U ⊂ C.

(i) p is called a removable singularity if f is bounded in a neighbourhood P n of p. The laurent serie f(z) = n cn(z−p) has no terms with negative exponent.

(ii) p is called a pole if f → ∞ as z → p. The laurent serie f(z) = P n n cn(z − p) has a finite number of terms with negative exponents.

(iii) Lastly p is called a essential singularity if f has no limit as z → p. The P n Lauret serie f(z) = n cn(z − p) has infinite number of terms with negative exponents.

Now we look at the concept of a surface. Let S be a Haussdorf space. S is called a surface if each s ∈ S has an open neighbourhood Ui such that there exists a homeomorphism Φi : Ui → Vi, where Vi ⊂ C is open. We call (Φi,Ui) a chart. An atlas is a set of charts A such that for each s ∈ S there exists (Φi,Ui) where s ∈ Ui. Now let (Φi,Ui), (Φj,Uj) be two charts at s ∈ S then the functions

−1 Φi ◦ Φj :Φj(Ui ∩ Uj) → Φi(Ui ∩ Uj) 8 Chapter 2. General Setting are called transition functions. We call an atlas analytic where all transition functions are analytic. We call two analytic atlases A, B compatible if all the transition functions of charts (Φ,U) ∈ A, (Ψ,V ) ∈ B are analytic. Such atlases form an equivalence class called a complex structure. A surface with a complex structure is called a Riemann surface.

Example 5 C with the single chart (Id : C → C, C).

Example 6 Σ, with the two charts (Id, C), (1/z :Σ\{0} → C, Σ\{0}). We see that transition function f : C\{0} → C\{0}, z 7→ 1/z is analytic.

n Example 7 Σ, with the two charts (w1, Σ\{∞}), (w2, Σ\{0}) where w1 = z 1 n and w2 = ( z ) . The transition function f : z 7→ 1/z is analytic on C \{0}. Further we see that those charts are not compatible with the charts in example 6 thus provides Σ with a different complex structure.

Now we will define some function types on Riemann surfaces. Let S be a Riemann surface. A function f : S → C is called analytic if the function −1 f ◦ Φ : Φ(U) → C is analytic for all charts Φ on S. Now let S1 and S2 be Riemann surfaces and Φ1 and Φ2 be charts of respective surface. Then a function f : S1 → S2 is called holomorphic if the function

−1 −1 Φ2 ◦ f ◦ Φ1 :Φ1(U1 ∩ f (U2)) → C (2.9)

−1 is analytic whenever U1 ∩ f (U2) 6= ∅. In a similar way we call a function f : S1 → S2 conformal if the function in equation 2.9 is conformal. Example 8 The M¨obiustransformations are holomorphic

Example 9 When we work with Σ we will study functions at ∞. A function f :Σ → Σ is conformal at z = ∞ if f(∞) 6= ∞ and f(1/z) is conformal at z = 0. If f(∞) = ∞ then f is conformal at z = ∞ if 1/f(1/z) is conformal at z = 0.

Let f : S → C be a holomorphic function, where S is a Riemann surface. A singular point is a point p ∈ S such that the point Φ(p) is a singularity to the function f ◦ Φ−1C → C. We say that p is a singularity of the same −1 type as Φ(p) is for f ◦ Φ . Similarly f : S1 → S2 has a singularity at p if the function in 2.9 has a singularity at Φ1(p). If f only has non-essential singularities we can replace C with Σ. In this case we call f meromorphic. Example 10 The M¨obiustransformations are meromorphic. More generally, a structure preserving map f : X → Y is called a morphism. 2.2. Surfaces 9

2.2.2 Automorphisms

Let S1,S2 be two surfaces. A function f : S1 → S2 is called a homeomor- phism if f is bijective, continuous and f −1 is continuous. f is called a local homeomorphism if it is open and continuous. Let S be a surface, then a conformal homeomorphism f : S → S is called an automorphism. We shall denote the group of automorphisms of a surface S Aut S.

Example 11 The automorphisms on C are the functions az +b where a, b ∈ C, a 6= 0.

Now we want to show that the M¨obiustransformations are the automor- phisms of Σ. We show that they are conformal by using the following theo- rem.

Theorem 12 An holomorphic function f is conformal at every point z0 for 0 which f (z0) 6= 0.

Theorem 13 All M¨obiustransformations are conformal on Σ.

Proof. Let T be a M¨obiustransformation, then T is conformal since

ad − bc T 0(z) = 6= 0 (2.10) |cz + d|2 for z 6= −d/c, ∞. The remain cases are:

(i) z = ∞,T (z) 6= ∞. In this case we have c 6= 0

a + bz S(z) = T (1/z) = c + dz 1 S0(0) = − 6= 0 c2 S is conformal at z = 0 so T is conformal at z = ∞

(ii) z = ∞,T (z) = ∞. In this case we have c = 0 and thus a 6= 0

c + dz U(z) = 1/T (1/z) = a + bz 1 U 0(0) = 6= 0 a2 10 Chapter 2. General Setting

U is conformal at z = ∞ so T is conformal at z = ∞

(iii) z = −d/c (6= ∞). In this case we have c 6= 0 cz + d V (z) = 1/T (z) = az + b V 0(−d/c) = −c2 6= 0 V is conformal at z = −d/c so T is conformal at z = −d/c. Theorem 14 Each conformal map f :Σ → Σ is an automorphism. Note that we do not require f to be bijective in theorem 14, it follows as a consequence. As automorphisms are conformal we see that on Σ they map circles to circles. Thus we conclude that Aut Σ is the group of M¨obiustransformation. This leads to the following identification Theorem 15 Aut Σ ∼= PSL(2, C). The functions az¯ + b T (z) = , a, b, c, d ∈ C, ad − bc 6= 0 (2.11) cz¯ + d are called anti-automorphisms of Σ. As the complex conjugation, z 7→ z¯, is anti-conformal and anti-automorphisms are a composition of an automor- phism and complex conjugation we see that anti-automorphisms are anti- conformal. The composition of two anti-automorphisms is an automorphisms and the composition of an anti-automorphism and an automorphism is an anti-automorphism. So the anti-automorphisms and automorphisms forms a group denoted by Aut(Σ) which has Aut(Σ) as a subgroup of index 2. Aut(Σ) is also known as the extended M¨obiusgroup.

2.3 The Hyperbolic Plane

In this section we will study the hyperbolic plane and the automorphisms on it. We start with constructing a model of the hyperbolic plane called the poincar´ehalf-plane model. Let H = {z ∈ Σ; Im(z) > 0} . We see that an automorphism of H is an automorphism of Σ such that R ∪ {∞} is preserved. So if T ∈ Aut H then b a T (0) = ∈ R ∪ {∞} T (∞) = ∈ R ∪ {∞} (2.12) d c That means a, c, b, d ∈ R as we want H mapped to itself. Thus we conclude that 2.3. The Hyperbolic Plane 11

Theorem 16 Aut H ∼= PSL(2, R). We define the hyperbolic length by the poincar´emetric dx2 + dy2 |dz|2 ds2 = = (2.13) y2 y2 Let γ be a piecewise differentiable path γ : [0, 1] → H, γ(t) = x(t) + y(t) = z(t) then the hyperbolic length of γ, h(γ), is given by q Z 1 dx 2 dy 2 Z 1 dz ( dt ) + ( dt ) | | h(γ) = dt = dt dt (2.14) 0 y 0 y Theorem 17 Let γ be defined as above, then T ∈ PSL(2, R) ⇒ h(T (γ)) = h(γ) Proof. Let az + b T (z) = ∈ PSL(2, R) cz + d then ad − bc 1 T 0(z) = = (cz + d)2 (cz + d)2 y dT Also if z = x + iy and T (z) = u + iv then v = |cz+d|2 so | dz | = v/y. So Z Z Z Z 1 | dT | 1 | dT dz | 1 v| dz | 1 | dz | h(T (γ)) = dt dt = dz dt dt = dt dt = dt dt = h(γ) 0 v 0 v 0 yv 0 y

Hyperbolic lines, or H-lines, in H are euclidian lines perpendicular to the real axis and semicircles centered at a point on the real axis. The following theorem motivates this definition.

Theorem 18 Let z0, z1 ∈ H then the path of shortest hyperbolic length from z0 to z1 is a segment of the unique H-line joining z0 and z1.

With this result we define the hyperbolic metric ρ as ρ(z0, z1) = h(γ), where γ is the H-line segment joining z0 and z1. ρ(z0, z1) is given by µ ¶ |z0 − z¯1| + |z0 − z1| ρ(z0, z1) = ln (2.15) |z0 − z¯1| − |z0 − z1| Example 19 Let a < b ∈ R then by formula 2.15 µ ¶ µ ¶ µ ¶ |ia + ib| + |ia − ib| 2b b ρ(ia, ib) = ln = ln = ln . |ia + ib| − |ia − ib| 2a a 12 Chapter 2. General Setting

H D

Figure 2.2: Hyperbolic lines in H and D.

H with the hyperbolic metric gives us a model for the hyperbolic plane, the poincar´ehalfplane model. Sometimes we will use another model, the poincar´edisc model D = {z ∈ C||z| < 1}. H can be mapped biholomorphi- z−i cally to D by z 7→ z+i as we have seen before in example 3. This induces a metric on D given by   z − i iw + i dz 2i 2|dw|  w = ⇔ z = ⇒ = ⇒ |dz| = z + i −w + 1 dw (1 − w)2 |1 − w|2 ⇒  1 − |w|2  y = |1 − w|2

2|dw| |dz| |1 − w|2 2|dw| ds = = = y 1 − |w|2 1 − |w|2 |1 − w|2 From now on we will call the hyperbolic plane H and the hyperbolic metric ρ regardless which of the models we use. Let E ⊆ H, then the hyperbolic area of E is defined as ZZ dxdy µ(E) = 2 (2.16) E y Theorem 20 Let E ⊆ H and T ∈ PSL(2, R) then µ(T (E)) = µ(E)

A hyperbolic n-sided polygon is a closed set in the closure of H in Σ bounded by n hyperbolic line segments. A point where two line segments intersect is called a vertex of the polygon. A vertex may be on R ∪ {∞}.A subset C of H is hyperbolically starlike if there is a point p in the interior of C such that ∀q ∈ C, the H-line joining p and q is in C. If this is true for all p ∈ C, C is called hyperbolically convex. 2.3. The Hyperbolic Plane 13

α β

γ

Figure 2.3: Hyperbolic triangle

We define the angle of two intersecting H-lines by the angle between their tangents at the point of intersection. If they intersect on R ∪ {∞} the angle is zero. The angles of a hyperbolic triangle are related to its area in a fascinating way.

Theorem 21 Gauss-Bonnet. Let ∆ be a hyperbolic triangle with angles α, β, γ. then µ(∆) = π − α − β − γ. (2.17)

Proof. We assume that ∆ has two sides that are vertical H-lines then the base is a segment of a semi-circle. As previously stated the area is invariant under transformations in PSL(2, R), thus we can assume that the semi-circle has center in 0 and radius 1. Assume that the vertical H-lines intersects the real axis at a, b where a < b. Let α, β be the angles at the vertices of ∆ were x = a and x = b respectively. Then γ = 0 since the last vertex is ∞. Now ZZ Z Z Z dxdy b ∞ dy b dx µ(∆) = = dx = √ = [x = cosθ] = 2 √ 2 2 E y a 1−x2 y a 1 − x Z β −sinθdθ = = π − α − β π−α sinθ If ∆ has one vertex c in R then there exists a T ∈ PSL(2, R) such that T (c) = ∞. As this transformation does not change the area we see that µ(∆) = π −α−β. Lastly if ∆ has no vertices in R∪{∞} we can construct ∆ as the difference of two hyperbolic triangles ∆1, ∆2 which has one common vertex on R ∪ {∞}. We can easily see that µ(∆) = µ(∆1) − µ(∆2) = π − α − β − γ The Gauss-Bonnet formula can be generalized to an arbitrary hyperboli- cally starlike polygon. 14 Chapter 2. General Setting

Theorem 22 Let Π be a n-sided hyperbolically starlike polygon with angles α1, α2 . . . αn. Then

µ(Π) = (n − 2)π − α1 − α2 · · · − αn (2.18)

2.4 Fuchsian groups

We have seen how PSL(2, C) can be divided in conjugacy classes. PSL(2, R) can be divided in conjugacy classes in a similar way, though some elements may be conjugate when regarded as elements of PSL(2, C) but not when regarded as elements of PSL(2, R). Let T ∈ PSL(2, R), as in 2.6 with R instead of C, then T is (i) parabolic if (|a + d| = 2). T has one fixed point α ∈ R ∪ {∞} and is conjugate to z + 1 or z − 1 (ii) hyperbolic if (|a + d| > 2). T has fixed points α, β ∈ R ∪ {∞} and is conjugate to λz (λ > 1) (iii) elliptic if (|a + d| < 2). T has fixed point ξ ∈ H conjugate to a rotation of H

Before looking at Fuchsian groups we need to define topological groups. A topological group is a topological space G which also is a group where the maps m : G × G → G m(g, h) = gh, i : G × G i(g) = g−1 are continuous. Example 23 PSL(2, C) and PSL(2, R) are topological groups. A Fuchsian group Γ is a subgroup of PSL(2, R) such that, when regarded as a subspace of PSL(2, R), it has the discrete topology. This means the set {g ∈ Γ; kgk ≤ k} is finite for every k > 0, where kgk is the matix norm. It is difficult to decide when a subgroup of PSL(2, R) is discrete. We shall look at a few examples.

Example 24 The simplest Fuchsian groups are the following cyclic groups: Hyperbolic cyclic groups are generated by a hyperpolic element, such as z 7→ λz, (λ > 0). Parabolic cyclic groups are generated by a parabolic element, such as z 7→ z + 1. Finite elliptic cyclic groups are generated by an elliptic element. 2.4. Fuchsian groups 15

Figure 2.4: Hyperbolic and parabolic elements.

Example 25 A more complicated group is the modular group PSL(2, Z) consisting of the following transformations az + b T (z) = a, b, c, d, ∈ Z ad − bc = 1. cz + d Now let Q be an H-line. An H-reflection is an isometry of H, other than identity, that fixes Q.

Example 26 The H-reflection in the imaginary axis Q0 is given by

R0 : z 7→ −z.¯

As for every H-line Q there exists a T such that T (Q) = Q0. So the H- −1 reflection in Q is given by T R0T . H-reflections are anti-conformal homeo- morphisms, that is they preserve angles but reverse orientation. We see that all H-reflections and other anti-conformal homeomorphisms of H are given by az¯ + b z 7→ ad − bc = −1. (2.19) cz¯ + d Example 27 As a last example let ∆ be a hyperbolic triangle with vertices v1, v2, v3 with angles π/m1, π/m2, π/m3 respectively and with opposing sides ∗ M1,M2,M3. We call the reflection in the line containing Mi Ri. Let Γ ∗ ∗ be the group generated by R1,R2,R3. If Γ = Γ ∩ PSL(2, R), then Γ = Γ ∪ ΓR1. Γ is a Fuchsian group called a triangle group and is generated by R1R2,R2R3,R3R1 where

m3 m1 m2 (R1R2) = (R2R3) = (R3R1) = Id. With presentation we have Γ = hx, y|xm3 = ym1 = (xy)m2 = Idi = hx, y, z|xm3 = ym1 = zm2 = xyz = Idi .

We will later see that a group Γ acting on H is discrete if and only if Γ acts properly discontinuous. Before we look at this property we will look at the concept of covering maps. 16 Chapter 2. General Setting

R1

R R 2π 2 1 m3

Figure 2.5: Triangle with reflections Chapter 3

Covering Maps

Let X,Y be topological spaces. A map p : Y → X is called a covering map (or covering) when each point x ∈ X has a neighbourhood V such that [ −1 p (V ) = Uα (3.1) α is a union of pairwise disjoint sets Uα, each mapped homeomorphically onto V . V is called a distinguished neighbourhood. Y is called the covering space of X and for each x ∈ X p−1(x) is called a fiber over x. Theorem 28 If X is connected and p : Y → X is a covering then each fiber p−1(x), x ∈ X has the same cardinal number, which is called the number of sheets (or leaves) of the covering. If |p−1| = n < ∞ we say that the covering is an n-sheeted covering. Example 29 Let Y be a discrete space with |Y | = n. Then p : X × Y → X defined as p(x, y) = x is a covering. It’s easy to see that |p−1(x)| = n, that is the covering is an n-sheeted covering. Example 30 Let p = eit. Then p : R → S1 is a covering map. We also see that each fiber p−1(x) is infinite. Example 31 Let h : X → Y be a homeomorphism. Then h is a 1-sheeted covering map.

3.1 Fundamental Groups

Let X,Y be topological spaces. Two functions f, g : X → Y are called homotopic if there exists a continuous map H : X × [0, 1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x ∈ X. H is called a homotopy between f and g.

Bartolini, 2006. 17 18 Chapter 3. Covering Maps

Example 32 Let f, g be two constant maps f, g : X → Y, f(x) = p, g(x) = q. f and g are homotopic if and only if p and q are in the same pathwise connected part of Y . If there exists a path γ such that γ(0) = p and γ(1) = q we can define a homotopy H between f and g by H(x, t) = γ(t) for all (x, t) ∈ X × [0, 1].

A particular case of homotopy is path homotopy. A path on a topological space X is a continuous function a :[s0, s1] → X. We call the initial point a(s0) and the final point a(s1) endpoints. A path a is closed if a(s0) = a(s1). Now we are interested in homotopies of paths with their endpoints fixed. Let a, b :[s0, s1] → X be paths with the same endpoints, we say that a, b are homotopic paths if there exists a homotopy H between a and b such that

H(s, 0) = a(s),H(s, 1) = b(s) H(s0, t) = a(s0) = b(s0) (3.2) H(s1, t) = a(s1) = b(s1) ∀s ∈ [s0, s1], ∀t ∈ [0, 1]

Path homotopy is a equivalence relation which we will denote a ∼= b. We call the equivalence class [a] of a path a a homotopy class. Let a, b :[s0, s1] → X be two paths such that a(s1) = b(s0). We then define the product ab :[s0, s1] → X by ½ s1−s0 a(2s − s0) if s ∈ [s0, 2 ] ab(s) = s1−s0 (3.3) b(2s − s1) if s ∈ [ 2 , s1]

It’s easy to see that ab is a path with endpoints a(s0), b(s1). we also need −1 the inverse of a path. We define the inverse of a path a by a (s) = a(s0 + s1 − s), s ∈ [s0, s1].

0 0 Theorem 33 Let a, b be two paths with a(s1) = b(s0). If a , b are paths such that a ∼= a0, b ∼= b0 then ab ∼= a0b0 and a−1 ∼= a0−1.

We now define [a][b] = [ab] and [a]−1 = [a−1]. By theorem 33 we see that neither depend on the choices of a, b. We are most interested in closed paths, or loops, that is paths a such that a(s0) = a(s1) = x. We call x the base point of a. The following theorem shows that the homotopy classes of loops based at a point x forms a group.

Theorem 34 Let a, b, c be loops based at x ∈ X. Let ex be that constant path at x. Then

−1 −1 (i) [a][a ] = [a ][a] = [ex] 3.1. Fundamental Groups 19

(ii) [ex][a] = [a][ex] = [a]

(iii) [a]([b][c]) = ([a][b])[c].

By theorem 34 we see that the homotopy classes of loops based at a point x0 ∈ X forms a group. We call this group the fundamental group of X and denote it by π1(X, x0).

Theorem 35 If x0, x1 belong to the same pathwise connected component of X then π1(X, x0) and π1(X, x1) are isomorphic.

It follows that if X is pathwise connected then π1(X, x0) = π1(X, x1) for all x0, x1 ∈ X. In this case we simply write π1(X).

Example 36 π1(Σ) = π1(C) = π1(H) = {Id}, that is the trivial group. In fact if X is a simply-connected space then π1(X) = {Id} and as we know Σ, C, H are all simply-connected.

b

a

Figure 3.1: A torus

Example 37 Let T be a torus. The torus is pathwise connected so we don’t need to choose a base point. Let a, b be the classes of loops one ”lap” around the meridian and longitude repectively. We note that each class c of loop on m1 m2 T can be expressed in terms of a, b, c = a + b . π1(T ) = ha, b|[a, b] = Idi where [a, b] = aba−1b−1 is the commutator. Similarly for a surface S of genus g we have π1(S) = ha1, b1, . . . , ag, bg|Π[ai, bi] = Idi.

Example 38 Let S be Σ with three points removed. Let a, b, c be classes of loops one ”lap” around each point respectively. As ab = c we can conclude that π1(S) = ha, b|−i. 20 Chapter 3. Covering Maps

b1 b2

a1 a2

Figure 3.2: A double torus

3.2 Group Actions on Surfaces

Let G be a group of homeomorphisms of a topological space X. Then G acts freely on X if each point x ∈ X has a neighbourhood V such that g(V ) ∩ V = ∅ for all g ∈ G. All groups we will look at do not satisfy this, sometimes we need a weaker condition. G acts properly discontinuously on X if each point p ∈ X has a neighbourhood V such that g(V ) ∩ V 6= ∅ for a finite number of g ∈ G. If G acts freely on X then it acts properly discontinuous. The quotient space of the orbits Gx, x ∈ X is denoted by X/G. the canonical projection p : X → X/G associates each x ∈ X with it’s orbit Gx. An open set A ⊂ X/G is a set such that p(A)−1 is open in X.

b

a a

b

Figure 3.3: A torus as a quotient space

Example 39 Let w1, w2 ∈ C, w1/w2 ∈/ R and

Λ = {z 7→ z + mw1 + nw2; m, n ∈ Z}. 3.2. Group Actions on Surfaces 21

The two paths a, b : [0, 1] → C/Λ defined by a(s) = sw1 and b(s) = sw2 are closed as a(0) = a(1) = 0 = b(0) = b(1). It’s easy to see that a, b are not homotopic to each other nor to a constant path. Now we see that π1(C/Λ) = h[a], [b]; [[a], [b]] = 1i. So C/Λ is a torus.

Now recall that a group G acts freely on a space X if each point x ∈ X has a neighbourhood U such that g(U) ∩ U = ∅ for all g ∈ G.

Theorem 40 Let G be a group of homeomorphisms acting on a space X then the following statements are equivalent

(i) G acts freely on X

(ii) The canonical projection p : X → X/G is a covering

(iii) p : X → X/G is locally injective

Proof. (i)⇒(ii): As G acts freely there exists a neighbourhood V for each x ∈ X such that g(V )∩V = ∅ for all g ∈ G. This means that g(V )∩h(V ) = ∅ for all g, h ∈ G and W = p(V ) is isomorphic to V . Now let Ug = g(V ) then [ −1 p (W ) = Ug. g∈G

That means p is a covering. (ii)⇒(i): As p : X → X/G is a covering for each open subset W ∈ X/G there exists disjoint open subsets Ui ∈ X such that p(Ui) = W which means no points x1, x2 ∈ Ui are in the same orbit. Thus g(Ui) ∩ Ui = ∅ and we conclude that G acts freely on X.

Example 41 From theorem 40 we see that p : C → C/Λ is a covering where Λ is the lattice in example 39.

Example 42 Let Γ be a Fuchsian group without elliptic elements. As no g ∈ Γ has any fixed point in H we can for each z ∈ H choose a neighbourhood V such that g(V )∩V = ∅, which means that Γ acts freely on H. From theorem 40 it follows that the canonical projection p : H → H/Γ is a covering.

Example 43 Fuchsian groups with elliptic elements acts properly discontin- uous on H.

In fact we have the following relation between discreteness and properly discontinuity. 22 Chapter 3. Covering Maps

Theorem 44 A subgroup Γ ⊂ PSL(2, R) is discrete if and only if it acts properly discontinuous on H.

Proof. Let Γ be properly discontinuous. Assume that Γ is not discrete. Then there exists elements Tn ∈ Γ such that Tn → Id as n → ∞. This means Tn(z) → z for all z ∈ H. Thus for all neighbourhoods V of z there exists an N such that Tn(z) ∈ V, n > N. This violates the definition of properly discontinuous and we conclude that Γ is discrete. Let p : X → Y be a covering. A homeomorphism f : X → X such that p ◦ f = p is called a deck transformation. The set of deck transformations of a covering p forms a group under composition.

Example 45 In example 30 we saw that p = eit is a covering map p : R → 1 S . Let fk(t) = t+2πk, then p◦fk = fk. That is, fk is a deck transformation. It’s easy to see that the group of deck transformations of p is isomorphic to Z.

Example 46 Now let p : C → C/Λ be the covering seen in example 41. The lattice Λ = {z 7→ z + mw1 + nw2; m, n ∈ Z} is the deck transformation group and is isomorphic to Z × Z = {(m, n)|m, n ∈ Z}.

Let f : Y → X be a continuous surjective map. If there for any path a :[s0, s1] → X and any point y ∈ Y such that f(y) = a(s0) there exists a patha ˜ :[s0, s1] → Y such that a(s0) = y and f ◦ a˜ = a we say that f has the path lifting property. If the patha ˜ is unique we say that f has the unique path lifting property. Let p : Y → X and y ∈ Y such that p(y) = x. By H(y) we denote the homomorphism p# : π1(Y, y) → π1(X, x).

Theorem 47 Let p : Y → X be a covering and a, b : I → X be paths with the same endpoints and a,˜ ˜b : I → Y be their liftings at a point y ∈ Y . Then a˜(1) = ˜b(1) if and only if [ab−1] ∈ H(y).

3.3 Universal Coverings

Let p : U → X be a covering. We call p a universal covering if U is simply connected, that is the fundamental group is trivial. We call U the universal cover of X. If X is simply connected it is its own universal cover. If p : U → X is a universal covering and p0 : Y → X is another covering then there exists a covering p00 : U → Y such that p0 ◦ p00 is a covering. For Riemann surfaces we have the following theorem. 3.3. Universal Coverings 23

Theorem 48 Uniformization Let p : U → S be a universal covering of a surface S. Then U is one of the following spaces:

(i) C

(ii) Σ

(iii) H

Example 49 As we have seen before there exists a covering p : C → T where T is a torus. That means C is the universal cover of T .

Example 50 Let Tg be a surface of genus g > 1. Then there exists a uni- versal covering p : H → Tg.

Example 51 In fact, H is the universal cover of all surfaces except the sphere Σ, wich is its own universal cover, the torus T and the plane C which both have C as their universal cover. 24 Chapter 3. Covering Maps Chapter 4

Riemann Surfaces as Orbifolds

In our work with surfaces in chapter 3 we have only looked att surfaces as topological spaces, but Riemann surfaces have a differential structure which we want to include. When we have worked with the hyperbolic plane H we have so far excluded the elliptic elements of PSL(2, R). In order to include them and the differential structure of Riemann surfaces we need some additional concepts namely orbifolds and branched coverings.

4.1 2-Orbifolds

A 2-orbifold O is a 2-dimensional space such that there exists an atlas of folding charts (Vi,Gi, Φi,Ui). Each chart consists of an open set Vi ∈ C, a finite cyclic or dihedral group Gi acting on V , an open set Ui ∈ O and a folding map Φi : Ui → Vi wich induces a homeomorphism Ui → Vi/Gi. The charts in an atlas satisfy [ Ui = O (4.1) i −1 −1 and if Φi (x) = Φj (y) then there exists neighbourhoods x ∈ Vx ⊂ Vi −1 and y ∈ Vy ⊂ Vj and the transition function Φj ◦ Φi : Vx → Vy is a diffeomorphism.

Example 52 A dihedral group is a group

­ 2 2 n ® Dn = a, b|a = b = (ab) = Id

A cyclic group is a group

n Cn = ha|a = Idi

Bartolini, 2006. 25 26 Chapter 4. Riemann Surfaces as Orbifolds

Let O be an orbifold. A singular point p is a point such that there exists a chart with a group Gi 6= {Id} stabilizing p. We call the set of such points the singular set of O. The singular set of O together with the stabilizers of the points in it are called the stratification of O. Here the only singular points we will consider are called cone points. A cone point is a singular point such that Gi is a cyclic group, in our case generated by an elliptic element in PSL(2, C). Let mi be the order of the cyclic groups Gi stabilizing the cone points pi then O has Euler characteristic µ ¶ X 1 χ(O) = 2 − 2g − 1 − (4.2) m i i

Example 53 Let all Gi = {Id}. Then O is a Riemann surface without singularities.

Example 54 In general, observe that the complex structure of a Riemann surface in page 7 provides the surface with an orbifold structure.

Example 55 The charts in example 7 provides Σ with an orbifold structure with folding charts which induces homeomorphisms w1 = z ∈ C/Cn and w2 = 1/z ∈ C/Cn. We see that 0 and ∞ are cone points.

4.2 Branched Coverings

Here we will construct branched coverings with coverings in mind. Let X and Y be topological spaces. A map p : Y → X is called a branched covering when each point x ∈ X has a neighbourhood V such that [ −1 p (V ) = Uα (4.3) α is a union of pairwise disjoint sets Uα, where Uα/Gα → V is a homeomor- phism, where Gα is a finite cyclic, possibly trivial, group. A point x ∈ X such that G 6= {Id} is called a branch point. Let x be a branch point, a point y such that p(y) = x is called a ramification point. Let B = {x1, . . . , xr}, the set p−1(B) ⊂ Y is called the ramification of the covering. Note that p : Y \ p−1(B) → X \ B is a covering.

Example 56 The coverings constructed in chapter 3 are branched coverings such that all Gα are trivial.

Example 57 Let S1,S2 be two Riemann surfaces. A surjective continuous function p : S1 → S2 is a branched covering. 4.2. Branched Coverings 27

In theorem 40 we saw a relation between groups of free homeomorphisms and coverings. Now we state a similar relation for properly discontinuous groups and branched coverings.

Theorem 58 Let X be a topological space and G be a group of homeomor- phisms acting on X then the following statements are equivalent:

(i) G acts properly discontinuous on X.

(ii) The canonical projection p : X → X/G is a branched covering

Note that we don’t have a statement about injectivity. Assume that G = Cn, that is a cyclic group of order n. If x ∈ X is fixed by G then p : X → X/G is n-to-one near x. A diffeomorphism f : Y → Y such that p ◦ f = p, where p : Y → X is a branched covering, is called a deck transformation. Let G be the deck transformation group of a covering p, we call p a regular covering if Y/G = X.

4.2.1 Universal Branched Coverings Let p : U → X be a branched covering of X. We call p a universal branched covering if U is simply connected. Now we have a theorem similar to theorem 48 for Riemann surfaces. This theorem is known as Poincar´esfirst theorem.

Theorem 59 Uniformization Let p : U → S be a universal covering of a Riemann surface S. Then U is one of the following spaces:

(i) C

(ii) Σ

(iii) H

A 2-orbifold covered by U is called a good orbifold, otherwise it’s called bad.

Example 60 The only bad Riemann surfaces not covered by U are:

(i) The ”Teardrop”, which is a sphere with one cone point.

(ii) The ”Football”, which is a sphere with two cone points with different order. 28 Chapter 4. Riemann Surfaces as Orbifolds

Example 61 With the orbifold structure of Riemann surfaces in mind we get a similar result as in example 51. H is the universal branched covering of all good Riemann surfaces except the sphere which has Σ as universal cover and the tori that has C as universal cover.

Now we define the fundamental group π¯1(S) of a Riemann surface S as the deck transformation group of its universal covering. In this case we have U/π¯1(S) = S. If S has no singular points thenπ ¯1(S) = π1(S) as defined in chapter 3.

c c

a b

a b

d d

Figure 4.1: The pillow case

Example 62 The orbifold with fundamental group ­ 2 2 2 2 ® π¯1(S) = x1, x2, x3, x4|x1 = x2 = x3 = x4 = Id . is given in the figure above. S is the sphere with four cone points of order 2. Example 63 Let S be a Riemann surface with genus 2 and with singular points pi with orders mi, i = 1 ... 3. Then the fundamental group is given by

m1 m2 m3 π¯1(S) = ha, b, c, d, x1, x2, x3, |x1 = x2 = x3 = x1x2x3[a, b][c, d] = Idi . 4.2. Branched Coverings 29

Example 64 In a similar way the fundamental group of a Riemann surface S with genus g and cone points pi with orders mi, 1 . . . r is given by

m1 mr π¯1(S) = ha1, b1, . . . , ag, bg, x1, . . . , xr|x1 = ··· = xr = ΠixiΠj[aj, bj] = Idi .

A branched covering p : Y → X is regular if and only ifπ ¯i(Y ) E π¯1(X), that is the fundamental group of Y is a normal subgroup of the fundamental group of X. The deck transformation group G is given byπ ¯1(X)/π¯1(Y ). Let p : Y → X be a branched covering and γ : [0, 1] → X be a closed path. Then the liftingsγ ˜ : [0, 1] → Y are all closed or all open. The deck transformation ¯ group is then given byπ ¯1(X)/π1(Y ) 30 Chapter 4. Riemann Surfaces as Orbifolds Chapter 5

Poincar´esTheorem

Now we will work with Fuchsian groups acting the hyperbolic plane H. Re- member that a Fuchsian group is a discrete subgroup of the automorphisms, PSL(2, R), of H.

5.1 Fundamental Domains

Let Γ be a Fuchsian group. F ⊂ H is a fundamental domain to Γ if F is a closed set such that S (i) T (F ) = H T ∈Γ

◦ ◦ ◦ (ii) F ∩ T (F ) = ∅ ∀T ∈ Γ \{Id} where F is the interior of F

(iii) µ(δF ) = 0 where δF is the boundary of F

A special kind of fundamental domain is constructed the following way. Let Γ be a Fuchsian group and p ∈ H be a point not fixed by any element in Γ. Then the Dirichlet domain of Γ centered at p is

Dp(Γ) = {z ∈ H| ρ(z, p) ≤ ρ(z, T (p)) ∀T ∈ Γ} (5.1)

Theorem 65 If p isn’t fixed by any element of Γ \{Id} then Dp(Γ) is con- nected.

A fundamental domain F of a Fuchsian group Γ is called locally finite if ∀p ∈ F ∃ neighbourhood V (p) such that V (p) ∩ T (F ) 6= ∅ for finitely many T ∈ Γ.

Bartolini, 2006. 31 32 Chapter 5. Poincar´esTheorem

b

2π π m2 m1 R1 a b

2π m3

π a m1

Figure 5.1: Fundamental domain for a triangle group

Example 66 Let Γ be a triangle group as seen in example 27. Let p be a point in the middle of the side Mi of the triangle ∆. Then Dp(Γ) = ∆∪Ri(∆) where Ri is the reflection in Mi.

Theorem 67 Let F1,F2 be fundamental domains for a Fuchsian group Γ. Then µ(F1) = µ(F2).

Proof. ¡ [ ¢ X µ(F1) = µ(f1 ∩ gF2 ) = µ(F1 ∩ gF2) g∈Γ g∈Γ X −1 = µ(g F1 ∩ F2) = µ(F2) g∈Γ

5.2 The Quotient Space H/Γ

Let F be a fundamental domain for the Fuchsian group Γ. Then we say that H/Γ is compact if F is a compact subset of H.

Theorem 68 If H/Γ is compact then Γ contains no parabolic elements.

Example 69 A Dirichlet domain F can be non-compact in several ways. First if it has an infinite number of sides it is non-compact. Otherwise it is non-compact if it has a vertex on R ∪ {∞}, called a parabolic vertex, or is bounded by a part of the real line. 5.2. The Quotient Space H/Γ 33

A Dirichlet domain is bounded by H-lines and possibly by sections of the real line. A point p ∈ H where such H-lines intersect is called a vertex of the Dirichlet domain. Let F be a Dirichlet domain for Γ and let u, v be vertices of F . We say u and v are congruent if there exists T ∈ Γ such that T (u) = v. Congruent vertices form an equivalence class, and those are called cycles. We are interested in vertices fixed by elliptic element. If a vertex of a cycle is fixed by an elliptic element so are all vertices in that cycle. Such a cycle is called an elliptic cycle and its vertices elliptic vertices. Let s be a side of a Dirichlet domain for a Fuchsian group Γ. If T ∈ Γ \{Id} and T (s) is a side of F then s and T (s) are called congruent sides. T (s) is a side of T (F ) and T (s) = F ∩T (F ). The sides of F fall into congruent pairs. Let Γ be a Fuchsian group with H/Γ compact. Then a Dirichlet domain F of Γ is compact and so has a finite number of sides. Therefore F has finitely vertices and elliptic cycles. So Γ has a finite number of elliptic generators with periods m1, m2 . . . , mr. If H/Γ is a Riemann surface with genus g and singular points p1, . . . pr with stabilizers Gi = Cmi the when say that Γ has the signature (g; m1, m2 . . . , mr). With the signature we get a generalization of Gauss-Bonnets theorem.

Theorem 70 (Gauss-Bonnet) Let Γ have signature (g; m1, m2 . . . , mr). If F is a fundamental domain for Γ then à µ ¶! Xr 1 µ(F ) = 2π (2g − 2) + 1 − . (5.2) m i=1 i

Theorem 71 If F is a locally finite fundamental domain for a Fuchsian group Γ then F/Γ is isomorphic to H/Γ.

Proof. Let p1 : H → H/Γ and p2 : F → F/Γ be the canonical projections and i : F → H be the inclusion map. We define θ : F/Γ → H/Γ as θ(p2(z)) = p1(z), z ∈ F . θ is bijective since if p2(z1) = p2(z2) then then there is a side pairing T ∈ Γ and p1(z1) = p1(z2). Now let V1 ⊂ H/Γ be open, then

−1 −1 −1 p2 (θ (V1)) = F ∩ p1 (V1)

−1 which is an open subset of F as p1 is continuous. This means θ (V1) is open and therefore θ is continuous. Now let V2 ⊂ F/Γ be open. There exists U ⊂ H such that

−1 p2 (V2) = F ∩ U, p2(F ∩ U) = V2. 34 Chapter 5. Poincar´esTheorem

Now let [ V = g(F ∩ U). g∈Γ Then

p1(V ) = p1(F ∩ U) = p1 ◦ i(F ∩ U)

= θ ◦ p2(F ∩ U) = θ(V2).

We want to show that θ(V2) is open. To do this it is sufficient to show that V is open since p1 is an open map. Let z ∈ V , as V is Γ-invariant we may assume that z ∈ F ∩ U As F is locally finite there exists a neighbourhood N of z such

N ∩ Ti(F ) 6= ∅, i = 1 . . . n.

−1 We suppose that z ∈ Ti(F ), i = 1 . . . n, then Ti (z) ∈ F and

−1 p2(Ti (z)) = p2(z) ∈ V2.

Thus z ∈ Ti(F ∩ U) and for sufficiently small radius of N we have \ N ⊂ Ti(F ∩ U) i and N ⊂ V .

5.3 Poincar´esTheorem

Theorem 72 (Poincar´e) Let P ∈ U be a polygon with a side pairing gen- erating a group Γ satisfying

(i) for each vertex x0 of P there are vertices x0, x1, . . . , xn of P and el- ementsS g0(= Id), g1, . . . , gn of Γ such that gi(Ni) are non-overlapping and gi(Ni) = B(x0, ε) where Ni = {y ∈ P |d(y, x0) < ε}.

(ii) each gi+1 = gigs where gs is a side pairing and gn+1 = Id.

Further there exists ε such that for each point p ∈ PB(p, ε) is in a union of images of P . Then Γ is a Fuchsian group Γ and P is a fundamental domain for Γ. 5.3. Poincar´esTheorem 35

However, this is difficult to prove as there are many cases to consider. We will look at the case where P is a hyperbolic triangle or a pair of hyperbolic triangles.

M3 v2 π π v1 m2 m1

M1 M2 π m3

v3

Figure 5.2: Hyperbolic triangle

Theorem 73 Let ∆ be a triangle with vertices v1, v2, v3 with angles π/m1, π/m2, π/m3 respectively and with opposing sides M1,M2,M3. Let Ri be the ∗ reflection in Mi, i = 1, 2, 3. The group Γ generated by the reflections Ri is discrete and ∆ is a fundamental domain for Γ∗. Further let Γ ⊂ Γ∗ be the conformal subgroup. then ∆ ∪ Ri(∆) is a fundamental domain for Γ.

Before proving the theorem we will outline the steps of the proof. First we create a space X with pairs (g, z) ∈ Γ∗ × ∆. Then we show that X is homeomorphic to H and we get a covering H → H/Γ∗. So by theorem 58 we get that Γ∗ is properly discontinuous and by theorem 44 Γ is discrete. 36 Chapter 5. Poincar´esTheorem

R1

R R 2π 2 1 m3

Figure 5.3: Hyperbolic triangle with reflections Proof. We will start with the pairs (g, z) ∈ Γ∗ × ∆. We can think of this as disjoint copies of ∆, (g, ∆). Let X be the space with elements hg, zi given by

◦ (i) {(g, z)} if z ∈ ∆

◦ (ii) {(g, z), (gRi, z)} if z ∈ Mi

mi+2−1 (iii) {(g, z), (gRi, z), (gRiRi+1, z) ..., (g(RiRi+1) , z)} if z = vi+2.

That is, we identify pairs (g1, z), (g2, z) if g1(z) = g2(z). Letg ˜ : X → X defined byg ˜ : hh, zi → hgh, zi. The group G ofg ˜ is isomorphic to Γ∗ since if ◦ g˜ = f˜ then if z ∈ ∆

hg, zi =g ˜ hId, zi = f˜hId, zi = hf, zi and so g = f. Now it’s easy to see that [ g˜ hId, ∆i = X (5.3) g˜∈G and ifg ˜ 6= f˜ then ­ ◦ ® ­ ◦ ® g˜ Id, ∆ ∩ f˜ Id, ∆ = ∅. (5.4) Let Ni = {z ∈ ∆|ρ(z, zi) < ε} (5.5) 5.3. Poincar´esTheorem 37

for some ε. Suppose that zi = v1. Then for sufficiently small ε

m1−1 N1 ∪ R3N1 ∪ R2R3N1 ∪· · · ∪ (R2R3) N1 = B(v1, ε).

◦ ◦ Similarly if zi ∈ Mj then Ni ∪ RjNi = B(zi, ε) and if zi ∈ ∆ then Ni = B(zi, ε). We note that Ni ⊂ B(zi, ε). Now let h : X → H be defined by h : hg, zi 7→ g(z). Let A ⊂ H be open. Then

h−1(A) = {hg, zi |g(z) ∈ A} which is open since for each zi such that g(zi) ∈ A there exists Ni, g(Ni) ⊂ B(g(zi), ε) ⊂ A and {hg, zi |z ∈ Ni} is open. That means h is continuous. Let B ⊂ X be open. Then

h(B) = {g(z)| hg, zi ∈ B} which is open since each {hg, zi |z ∈ Ni} maps to B(zi, ε) where Ni is as a above. The last two results mean that h is a local homeomorphism. Now we will look at the bijectivity. Assume that h hg1, z1i = h hg2, z2i. then

−1 g1(z1) = g2(z2) ⇒ z1 = g1 g2(z2).

◦ −1 It’s easy to see that z1 = z2. If z1 ∈ ∆ then g1 g2 = Id and g1 = g2. If ◦ z1 ∈ Mi then g1 = g2 or g1 = g2Ri and similarly when z1 = vi. Thus we conclude that hg1, z1i = hg2, z2i, that is h is injective. Now let w ∈ H. Then there exists a path γ such that γ(0) = z ∈ ∆ and γ(1) = w. We note that we can lift γ(t) ∈ ∆ to a pathγ ˜(t) ∈ X by

γ˜(t) = hId, γ(t)i .

As hId, γ(t)i = hRi, γ(t)i if γ(t) ∈ Mi we see that we can continue this lifting as γ˜(t) = hRi,Ri(γ(t))i ∗ if γ(t) ∈ Ri(∆). Continuing this way we see that there exist g ∈ Γ such thatγ ˜(1) = hg, zi and g(z) = γ(t) = w. Thus we can conclude that h is surjective. We have shown that h is a homeomorphism and with equations 5.3 and 5.4 we see that [ g(∆) = H (5.6) g∈Γ∗ ◦ ◦ g(∆) ∩ f(∆) = ∅. (5.7) 38 Chapter 5. Poincar´esTheorem

w

z0

Figure 5.4: Path lifting with triangles

Now let p : H → H/Γ∗ be the canonical projection. For each point z ∈ H/Γ∗ there is a neighbourhood N constructed as above such that [ −1 h (N) = B(zi, ε) i where zi are the points in the orbit of z. For sufficiently small ε the open sets B(zi, ε) are pairwise disjoint. Each B(zi, ε)/Λ is homeomorphic to N where Λ is the stabilizer of z in Γ∗. Thus p is a branched covering and by theorem 58 Γ∗ is properly discontinuous and by theorem 44 discrete. Now we finally look at Γ ⊂ Γ∗, the conformal subgroup. Γ is discrete as ∗ Γ is discrete. Thus we only have to show that ∆ ∪ Ri(∆) is a fundamental ∗ domain for Γ. Let g ∈ Γ be anti-conformal. Then gRi is conformal and thus gRi = T ∈ Γ which means g = TRi. It follows that [ H = g(∆) = g∈Γ∗ ¡ [ ¢ ¡ [ ¢ T (∆) ∪ T (Ri(∆)) = T ∈Γ T ∈Γ [ T (∆ ∪ Ri(∆)). T ∈Γ

◦ ◦ Further it is easy to see that if T 6= S then T (F ) ∩ S(F ) = ∅, where F = ∆ ∪ Ri(∆). We conclude that Γ is a Fuchsian group and ∆ ∪ Ri(∆) is a fundamental domain for Γ. With Poincar´estheorem we get the following theorem. 5.3. Poincar´esTheorem 39

Theorem 74 If g ≥ 0 and mi ≥ 2 are integers and if µ ¶ Xr 1 2g − 2 + 1 − > 0 (5.8) m i=1 i then there exists a Fuchsian group Γ with signature (g; m1, m2 . . . , mr).

We will end this chapter with a few examples.

Example 75 Let P be the polygon with side pairing we get with the union of triangles and with the side pairing in figure 5.5. We see that all the vertices which are endpoints to the sides ai, bi are congruent and with angle sum 2π. We can then scale the polygon such that à µ ¶! Xr 1 µ(P ) = 2π 2g − 2 + 1 − . m i=1 i We note that the cycle conditions are satisfied and by theorem 72 P is a fun- damental domain and the group Γ generated by the side pairing is a Fuchsian group. With presentation we have

m1 mr Γ = ha1, b1, . . . , ag, bg, x1, . . . , xr|x1 = ··· = xr = ΠixiΠj[aj, bj] = Idi . 40 Chapter 5. Poincar´esTheorem

x1 x1

x2

x2 a1

x3

b1 x3

x4

a1

x4

b1 b2

a2 a2

b2

Figure 5.5: Fundamental polygon 5.3. Poincar´esTheorem 41

Figure 5.6: The poincar´edisc tesselated by the modular group [4] 42 Chapter 5. Poincar´esTheorem

Figure 5.7: The poincar´edisc tesselated by the (2, 3, 7) triangle group [4] Conclusions and Further work

We have seen how Riemann surfaces can be expressed as quotient spaces with the use of discrete groups. We have also seen how hyperbolic triangles with side reflections generate discrete groups. The procedure we have used can easily be used for spherical or euclidian triangles as well. It is also possible to extend it to arbitrary starlike polygons. Further the theorem has many applications. It can be used as a tool to find automorphisms of a surface, in the work with Teichm¨ullerspaces and in many other applications.

Bartolini, 2006. 43 44 Chapter 5. Poincar´esTheorem Bibliography

[1] A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91 Springer-Verlag (1983)

[2] G. A. Jones, D. Singerman, Complex Functions, an algebraic and geo- metric viewpoint, Cambridge University Press (1997)

[3] E. L. Lima, Fundamental Groups and Covering Spaces, A K Peders, Ltd. (2003)

[4] W. Magnus Non-Euclidian Tesselations and Their Groups, Academic Press (1974)

[5] B. Maskit On Poincar´esTheorem for Fundamental Polygons, Advances in Mathematics 7, (1971) 219-230

[6] Y. Matsumoto, J. M. Montesinos-Amilibia, A Proof of Thurston’s Uni- formization Theorem, Tokyo J. Math. Vol. 14, No. 1 (1991) 181-186

[7] J. Milnor, On the 3-dimensional Brieskorn manifolds M(p,q,r), In Knots, groups and 3-manifolds. (Ed. L. P. Newwirth), Princeton Uni- versity Press (1975) 175-181

[8] E. B. Saff, A. D. Snider, Fundamentals of Complex Analysis with Appli- cations to Engineering and Science, Pearson Education, Inc. (2003)

Bartolini, 2006. 45 46 Bibliography LINKÖPING UNIVERSITY ELECTRONIC PRESS

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