Aseguramiento de la calidad 105

We can equip the group( PSL2(R) with a topology inherited fromR4 by identifying the matrix a b

c d )

with the vector (a,b,c,d)∈R4, then defining the norm on PSL2(R) to be the Eu- clidean norm onR4. This norm then induces a metric, which in turn induces the metric topology. Recall that a setE in a topological space(X,τ)isdiscreteif for eache∈E there exists an open subsetG∈τ such thatE∩G={e}. We make the following definition.

Definition 6.1.20. Let Gbe a subgroup ofPSL2(R). ThenG is said to be aFuchsian group if and only ifGis a discrete subset of the topological spacePSL2(R).

Another way to describe a Fuchsian group G is in terms of properly discontinuous group actions. We say that a groupGactsproperly discontinuouslyon a metric spaceX if and only if the orbitG(x):={g(x):g∈G}is locally finite for allx∈X. That is, given an orbitG(x), every compact subset K ⊂X contains at most finitely many points of G(x). Note that the statement that a group acts properly discontinuously is equivalent to the statement that each orbit ofGis a discrete set of points.

Proposition 6.1.21. Let G be a subset of Con(1). Then G is Fuchsian group if and only if G acts properly discontinuously onD2.

Proof. See Theorem 5.3.2 in [7].

Definition 6.1.22. LetGbe a Fuchsian group. Afundamental domain F forGis an open subset ofD2such that the following conditions are satisfied.

1. ∪

g∈G

g(F) =D2,

2. g(F)∩h(F) =/0, for allg,h∈Gwithg̸=h.

Thus, each fundamental domain for a Fuchsian groupGgives rise to atessellationof hyper- bolic space. Let us now describe a particular type of fundamental domain. LetGbe a Fuchsian group acting onD2and letz0∈D2be a point that is not fixed be any elliptic element of the group G. Then theDirichlet fundamental domain Dz0(G)ofGat the pointz0is given by

Dz0(G):={z∈D

2_:d

Alternatively, for each g∈G/{id}consider the perpendicular bisector of the geodesic segment joiningz0 andg(z0). This divides D2into two half-spaces. With Hgreferring to the half-space

containingz0, we have that

Dz0(G) =

∩ g∈G/{id}

Hg.

ThatDz0 really is a fundamental domain requires proof, but we leave this to the reader. As usual,

this definition could equally well have been written in terms ofH.

Example 6.1.23. Consider the group PSL2(Z):= {( a b c d ) :a,b,c,d,∈Zandad−bc=1 } /{±I}.

This group is referred to as themodular group. The modular group is generated by one parabolic elementPand one elliptic elementQ, where

P(z) =z+1 and Q(z) =−1 z.

It is clear that this is a discrete subgroup ofPSL2(R). It can be shown that the region bounded by the lines Re(z) =1/2, Re(z) =−1/2 and the unit circle is the Dirichlet fundamental domain at the pointz0=2ifor the modular group. (A proof of this fact is contained in Appendix B.)

We can use the notion of a Dirichlet fundamental domain to obtain an important theorem for Fuchsian groups. In the following discussion, letDz0(G)refer to a Dirichlet fundamental domain

for a Fuchsian group G constructed at the base point z0. The region Dz0(G) is a hyperbolic

polygon inD2∪S1(in a wider sense than is usual, since we allow vertices and edges onS1 and allow the possibility that there be infinitely many edges) . Let us consider the edges which bound the polygonDz0(G), where we letsgdenote the edge that is part of the perpendicular bisector of

the segment joiningz0tog(z0). Observe that

z∈sg ⇔ dh(z,z0) =dh(z,g(z0)) ⇔ dh(g−1(z),g−1(z0)) =dh(g−1(z),z0) ⇔ g−1(z)∈sg−1.

In other words, we have that for the edgessgwhich bound the polygonDz0(G)we have that

g−1(sg) =sg−1 and g(s_g−1) =s_g.

We refer to these identifications of edges under elements ofGas theside-pairing transformations ofG. Note that ifgis a parabolic element that is also a side-pairing transformation, then the sides that are paired bygwill meet at the fixed point ofg.

Theorem 6.1.24. The side-pairing transformations of a Fuchsian group G for a Dirichlet funda- mental domain Dz0(G)are generators of the group G.

Definition 6.1.25. A Fuchsian groupGis said to be geometrically finite if there exists a funda- mental domain forGwith only finitely many edges.

From the example given above of a fundamental region for the modular group, it is apparent that the modular group is geometrically finite.

Remark 6.1.26. A Fuchsian groupGis geometrically finite if and only ifGis finitely generated. This can be deduced immediately from Theorem 6.1.24. (The equivalence no longer holds for discrete groups of isometries of higher dimensional hyperbolic spaces.)

Recall that a Riemann surfaceis a connected, analytic, complex 1-dimensional manifold. A Riemann surfaceSis calledsimply connectedif every closed curve onScan be continuously de- formed into a single point (so the surface of the 2-sphere is simply connected, whereas the torus is not). It is a very deep theorem in the theory of complex functions - the Riemann Mapping Theo- rem, sometimes called the First Uniformization Theorem - that every simply connected Riemann surface is conformally equivalent to one ofC, C∪ {∞}orD2. Further, the Second Uniformiza- tion Theorem states that every Riemann surfaceSis conformally equivalent to a quotient ˜S/Gfor some simply connected Riemann surface ˜Sand for some groupGof conformal automorphisms which acts properly discontinuously on ˜S. The quotient ˜S/Gcomprises equivalence classes of points in ˜S, where two points are equivalent if and only if they belong to the sameG-orbit. If we are in the case where ˜Sis conformally equivalent toD2, then every properly discontinuous group Gis a Fuchsian group. So, here we always have that a Riemann surface conformally equivalent toD2/G is represented by a fundamental domain for the action ofG. We can also think of this the other way around - that every Fuchsian groupGhas an associated Riemann surface, obtained by “gluing” the edges of a fundamental domainF forG.

In the figures below, we illustrate various types of surfaces obtainable as the Riemann surface associated to a Fuchsian group. Figure 6.2 shows a compact surface, which occurs when the fundamental domain of the groupGdoes not have any vertices onS1and also shows an example of a surface with funnels. This happens when the fundamental domain has edges contained in S1_{. Finally, going back to the example of the modular group, in Figure 6.3 we see the modular} surface. This surface has what is known as acusp, which happens when the groupGcontains a parabolic element. There is then a parabolic fixed point as a vertex of the fundamental domain.