## Series C. Spaces of Kleinian groups

Panagiotis Laganas. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. Four circles to the kissing come. The smaller are the benter. The bend is just the inverse of The distance from the center. To spy out spherical affairs An oscular surveyor Might find the task laborious, The sphere is much the gayer, And now besides the pair of pairs A fifth sphere in the kissing shares.

Yet, signs and zero as before, For each to kiss the other four The square of the sum of all five bends Is thrice the sum of their squares. Initially an account of hyperbolic geometry in dimensions two and three is introduced, mentioning basic properties and models.

We will then examine the group of orientation- preserving isometries of hyperbolic space, concentrating on discrete subgroups, namely Kleinian groups, and their limit sets. Furthermore, we attempt to explore Apollonian circle packings and their relation with hyperbolic geometry.

Finally, we include a significant programming element at the end of the study, while the output images are used throughout. Chapter 1 Introduction We are all familiar with Euclidean geometry, since it resembles the world we live in, at least locally in space. On the contrary, we are least acquainted with non-Euclidean geometry and in particular hy- perbolic geometry. Hyperbolic geometry has been an interesting subject of active re- search ever since.

This is mainly because it relates to a number of diverse areas of mathematics, including but not limited to Complex Analysis, Group Theory and Low-dimensional topology. Therefore, we could suggest that its development was not only based on interest but also necessity. In the current study we follow a path of these connections, which gradually lead us to our focal point, the Apollonian gasket.

We start our exploration in chapter two with an account on three commonly used geometric models for hyperbolic geometry and their properties. In the end of this chapter we study and classify the isometries of hyperbolic space and display their relation to linear fractional transformations on the extended complex plane. In the third chapter we realise the topological group structure of hyperbolic isometry groups. Thereafter, we concentrate on discrete groups of hyperbolic isometries, namely Kleinian groups, and the prop- erties of their limit sets.

These were produced using our own algo- rithms based on the pseudocodes from [Wri02]. Last, an attempt to shortly explore the notion of Hausdorff dimension was made. Apart from the algebraic con- struction, we display two different but closely related Kleinian groups, namely the Apollonian group and a specific Schottky group, that give rise to the same Apollonian packing.

In addition, we remark on their hidden relations. In the end comments are put forth on a convergence relation that appears between the aforementioned Schottky group and a set of special Kleinian groups. Chapter 2 Hyperbolic geometry 2. They are distinguished by having zero, constant positive or constant negative sectional curvature. These are called Euclidean, spherical and hyperbolic geometry, respectively. The aim of this chapter is to study the structure of hyperbolic geometry.

In order to better understand different aspects of this geometry we equip ourselves with various geometric models. Models of hyperbolic space, denoted by Hn , are comprised of an underlying space along with a metric.

## Discontinuous Groups and Riemann Surfaces (AM-79), Volume 79

With the help of these we can then define geodesics, see shapes or compare areas. Furthermore, we examine the isometries of hyperbolic space and categorize them in conjugacy classes. Although all three models could serve as models of n-dimensional hyperbolic geometry, we shall focus on dimensions two and three.

These are also given a special name, the boundary at infinity. Points that lie on the boundary at infinity are called ideal points or points at infinity. This is a conformal model for hyperbolic space, in other words angles in the actual space are preserved in the model. Figure 2. Hyperbolic lines are either semi-circles perpendicular to the real axis or Euclidean straight lines ending on the real axis.

Similarly, in the case of U3 hyperbolic lines are represented by semi-circles or lines orthogonal to the boundary at infinity. Some examples in U2 are given in Figure 2. Then, we can consider vertical lines as circles with infinite radius. It is obvious that two hyperbolic lines meet in at most one point. As discussed in Section 2. These hy- perbolic lines are the parallel lines of Un. Imz 2 Remark: The name boundary at infinity comes from the fact that as we approach the boundary of Un the distance becomes infinite.

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In order to obtain a formula for the general distance between two points in Un along a geodesic, we utilize the cross-ratio. Suppose we have two points x, y in U2 then from our knowledge in Euclidean geometry there is a unique circle or line passing through x, y and meeting the boundary; that would be the hyperbolic geodesic connecting those two points.

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• We omit the calculations due to their extensiveness. Theorem 2. Angles are evaluated in the usual way as in Euclidean space; that is by measuring the angle of the tangents at the intersecting points. In the previous example there were two ideal vertices and hence the corre- sponding angles were 0, whereas the third one was clearly a right angle. Hyperbolic triangles with all three vertices at infinity are called ideal.

The following Gauss-Bonnet theorem shows how the area of a hyper- bolic triangle is fully determined by its angles.

As an application of the Gauss-Bonnet theorem we can find the area of the hyperbolic triangle in Figure 2. One of the benefits of this model is that we can view the whole hyperbolic space on its interior, since the n-dimensional unit disc is bounded in Euclidean space.

Examples of geodesics are demonstrated in Figure 2. Similar to the discussion in the previous section we deduce that if two lines in Bn are disjoint then they are parallel. Furthermore, there exists a conformal mapping from the upper half-space Un to the unit ball Bn according to the Riemann mapping theorem [Sin87]. As an example, we can see in Figure 2. That is also one of the advantages of this model.

Using the infinitesimal metric of Bn we can also derive a simple formula for the hyperbolic distance of a point z in Bn from its centre O. However, the metric and geodesics are not the same. In order to avoid any confusion we will denote it by Dn. Euclidean straight line segments in Dn with endpoints on the sphere at infinity represent hyperbolic lines, as shown in Figure 2. Unlike the previous models, the Klein disc model is not conformal, that is it distorts angles. Nevertheless, it is beneficial in the sense that we can handle problems in hyperbolic geometry as in Euclidean.

For instance, we could show properties of incidence and convexity for subsets of Hn [Bea83]. Definition 2. Similar to Section 2.

The metric dD on Dn is given by! We can then use the functions that take one model to the other, as described in the previous sections, to derive the group of isometries in the rest of the models.

## Free Series C. Spaces Of Kleinian Groups

Below we display some of their basic properties which will prove useful for our study. We now classify the isometries of hyperbolic space in terms of their fixed points. Analogously, the same classification is used for the elements of P SL 2, R with the exception of loxodromic transformations. Now, taking into ac- count that the trace of a matrix is also invariant under conjugation we are able to classify the isometries with regard to the function trace2.

Examples from each class are illustrated in Figure 2. Chapter 3 Discrete isometry groups and limit sets 3. Moreover, we introduce other related concepts such as discontinuity, limit sets and ordinary sets.

## Tilings of the hyperbolic space and their visualization

Apart from presenting their individual attributes, we make an attempt to explore the connections between them. Distinguished cases of discrete groups are also intro- duced, Kleinian, Fuchsian and Schottky groups.

Finally, an account on Hausdorff or fractal dimension is included in the last section. Before moving on, we remind the reader some basic notions of group theory. Recall that SL 2, C is a topological group with the Euclidean metric topology. Similarly, this result holds for P SL 2, R.