Structure of the Modular Group

We structure our investigation of the modular group by considering four cyclic subgroups. Recall that a cyclic subgroup can be generated by computing all powers of a single group element. The four cyclic subgroups we present are representative of the four possible types of subgroups found in the modular group.

Examples of Cyclic Subgroups

For our first subgroup, consider the function , that is, a Möbius transformation with coefficients , , , . We subscript the function because is an element of order two. Thus, is its own inverse, so that . In this case generates a subgroup with only two elements, namely, . Note that in this article we adopt the standard notation that parentheses ( ) indicate the set of elements generated by taking all powers of the elements enclosed by the parentheses. Curly braces { }, on the other hand, enclose the delineated list of elements in the set.

Figure 6 depicts the way in which maps the two fundamental regions, shown in Figure 4, onto one another. In fact, the action of on the fundamental regions is to "rotate" them onto each other about the central fixed point . The actual mapping is performed in a single step. In particular, only the first and middle frames contain illustrations of fundamental regions. However, we have defined a sequence of intermediate mappings in order to illustrate through animation the mapping properties of . In a later section we discuss how the function was broken into parts in such a way that the hyperbolic nature of the motion was preserved.

Figure 6. Action of an order-two element, .

This example highlights the fact that vertical lines are paths of least distance in the upper half-plane model of the hyperbolic plane. Indeed, it is usual to view straight lines as circles that have radii with infinite length and that pass through the point at infinity. With this "bending" of the definition of circles, vertical lines have all the characteristics required of geodesics in the hyperbolic plane. They are perpendicular to the x-axis that is the boundary of the upper half-plane model of the hyperbolic plane, and they are circles that pass through the point at infinity.

Our second example is a subgroup of infinite order generated by the linear shift , that is, a Möbius transformation with The function has infinite order, for and no point is ever returned to its original position no matter how many times is applied. The subgroup produced by taking all powers of and its inverse, is denoted as . The hyperbolic isometry is notable among the elements of the modular group because it is also a Euclidean isometry.

Every point in the plane shifts one unit to the right under the action of . The infinite half strips in the following animation are images of each other under powers of . For contrast, we also provide images of these infinite half-strip regions under the map . Under the hyperbolic metric, the "smaller" regions in red are each congruent to the half-strip regions in blue. These images are bunched in a flower-like arrangement attached to the real axis at the origin. As the blue infinite regions are pushed from left to right, their red images echo their motion in a counterclockwise direction. Note that these two actions are not produced by a single transformation. As discussed in a later section, the two transformations that cause these actions are closely related to each other as algebraic conjugates.

Figure 7. Copies of fundamental regions moving back and forth with corresponding regions anchored at the origin.

The third cyclic subgroup of that we consider is generated by a composition of the first two functions. We define . The subgroup generated by this element is denoted , a subgroup of order three. We viewed the action of this subgroup in Figure 2. A number of fundamental regions are shown associated with the blue fundamental region attached to the origin in the animation's first frame. We include these in order to provide a better orientation for the scene. A brief pause is inserted at the end of each complete action of , a rotation one-third the way around the fixed point, . Of special interest is how the common point of the blue cluster moves as the rotation takes place. The point begins at the origin and slides towards the left along the negative x-axis. When the blue lines of the cluster become vertical, this is precisely when that common point arrives at or passes through the point at infinity! All of the attached blue curves then become parallel lines that "meet at infinity." The point then slides along the positive x-axis to arrive once again and pass through the origin. It is fair to say that the motion of the point, as it passes through the origin, is a "mirror image" of the motion of the point as it passes through the point at infinity.

Figure 8. Action of an order-three element, .

The rotations we saw in the action of and are of order two and order three, that is, after the rotation is repeated a number of times, all points are back to their original positions. In contrast, the function generates an infinite subgroup. When we iterate , the right shifts accumulate on the point at infinity. Points in the left half-plane get repelled by infinity, while points in the right half-plane get attracted to infinity. Of course, since all points in the left half-plane eventually map to points in the right half-plane, all points are, in some sense, simultaneously attracted to and repelled by infinity under the action of . Indeed, the point at infinity is the single fixed point for the action of . In the final section of this article we will produce a fixed point associated with the action of that we can actually see.

There is another type of transformation in the modular group that generates an infinite subgroup . This subgroup is different from in the sense that there are two distinct fixed points--one an attractor, the other a repeller. The generator for the subgroup of our example is . The following animation depicts the action of on fundamental regions in the plane. Note that all points exterior to the red circle on the left are mapped to the interior of the green circle on the right. The variety of colors outlining fundamental regions makes it easier to follow individual regions as the transformation progresses. The animation begins with regions exterior to the red and green circles. These regions are all mapped to the area between the green circle and smaller yellow circle. If the action of were to be repeated, the regions would be mapped into the interior of increasingly small circles inside the smallest (yellow) circle shown. The attracting fixed point for lies within these shrinking, nested circles.

Figure 9. Action of a hyperbolic element, .

The rotations and translations we have seen as examples are intimately related to Euclidean rotations and translations (see Coding Considerations). The transformation is related in a similar way to a Euclidean dilation. Recall that Euclidean dilations produce figures that are similar, but not congruent. A curious characteristic of hyperbolic space is that such distinctions disappear in the hyperbolic plane. It is sufficient for figures to have the same angles to guarantee congruence. In marked contrast to Euclidean space, equal angles guarantee that corresponding side lengths are equal in the hyperbolic metric.

Generators of M

The entire group can be generated by the two functions and , that is, . Establishing this fact requires tools from linear algebra about which we will make only a few brief comments. The group of matrices with real number entries and with nonzero determinants has been studied extensively and much is known about it and its subgroups. While the modular group cannot be represented as a subgroup of , it "almost" can be. For instance, the elements and are considered two distinct elements in ; however, the actions of the two associated Möbius transformation are identical because . In general, for every element in the modular group there will be two elements in associated with it. A remarkable feature of Möbius transformations is that the group operation of compositions produces coefficients that are identical to the results of matrix multiplication. To see this, consider the two Möbius transformations and . We first multiply the associated matrices:

Next we carry out the composition of the two functions:

The coefficients and the four matrix entries are the same!

In this way the group operation of composition of functions in the modular group can be "replaced" with the group operation of matrix multiplication in . It is down this path we would travel if we were to present a complete proof of the claim that the modular group is generated by the two elements and .

A major part of this claim is that any element of can be written as a composition of and . Consider the following examples of compositions.

Note that both of these functions have a coefficient matrix with determinant equal to one. A worthy exercise for undergraduate mathematics students is to verify by direct computations that each of the equalities hold for the indicated compositions.