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The Modular Group

# Introduction

Transformations of spaces have long been objects of study as evidenced by many of the early examples of groups. Among the most important transformations are the isometries, those transformations that preserve lengths. Euclidean isometries are translations, rotations, and reflections. The groups and subgroups of Euclidean isometries of the plane are so familiar to us that we may not think of them as revealing much about the space they transform. In hyperbolic space, however, light traveling or even a person traveling on a hyperbolic shortest-distance path will tend to veer away from the boundary. Thus, the geometry is unusual enough that viewing the actions of isometries of hyperbolic 2-space reveals some of the shape of that space. Two-dimensional hyperbolic space is referred to as the hyperbolic plane.

Figure 1. The upper half-plane model of the hyperbolic plane.

We will be examining how elements of the modular group rearrange the triangular-shaped regions shown in the previous diagram. The curved paths are arcs of circles orthogonal to the x-axis. In Euclidean spaces, the shortest-distance paths are straight lines. In hyperbolic space, shortest paths lie on circles that intersect the boundary of the space at right angles. Hyperbolic distances are computed in a way so that there is a "penalty" to be paid for traveling near to the plane's boundary. Thus, the shortest-distance paths (or hyperbolic geodesics) between two points must "bend" away from the boundary.

In the animations that follow, it will be instructive to focus on the action that a transformation takes on the family of circles that meet the x-axis at right angles. The transformations that we consider, namely, members of the modular group, preserve this family of circles. The circles in the family are shuffled onto different members in the family, but no new circles are created and none taken away. One could say that, in the context of hyperbolic geometry, the transformations preserve the family of all shortest-distance paths. Indeed, this is an excellent thing for an isometry transformation to do.

The following animation is an example of how a single element of the modular group acts on a cluster of regions, moving them about a common fixed point. This action will be discussed in detail in a later section. Note that increasing the integer input for the corresponding animation function will produce more frames and a smoother animation, but will also result in a longer waiting time while the frames are being created.

Figure 2. A single element of the modular group acting repeatedly on a cluster of regions.

Fuzzy, pixilated screen images reveal, in their own way, more than the crisp, still shots that we can obtain from the modern laser printer. The dynamic representation of the relationships yields an intuitive understanding that can easily exceed that from any number of crisp, still shots. If a picture is worth a thousand words, then, even with limited screen resolution, an animation is worth a thousand pictures.

The context of this paper is described in Chapter 2 of Ford's Automorphic Functions [1]. In this small text one can find illustrations that inspired our animations. The formulas, which made coding the animations much simpler than one might expect, are given and justified in detail.