Hitting the edge

Figure 8.14 shows two pictures in which one of the two traces is 2 and the other is 3. In other words, one of the two generators a and b is parabolic but the other is not. Both pictures are rather like the gasket picture in frame (vi) of Figure 8.2, but on the left only the circles Ca and CA have come together with an extra point of tangency, while on the right the tangency is between the circles Cb and CB. This may be easier to see if you compare the left frame of Figure 8.14 to Figure 8.4. See how the fixed points of a have come together pinching off the lefthand part of the picture from the right. If, on the other hand, we fix ta and send tb to 2, the upper and lower pincers come together resulting in the righthand frame of Figure 8.14.

The myriad small circles in these pictures appear for exactly the same reason as they did in the last chapter. If for example a is parabolic, then so is bAB, and so also is abAB. Thus the two elements a,bAB generate a subgroup conjugate to the modular group, which as we know means we expect to see circles in the limit set. Well, here they are!

Groups like these in which one element is parabolic are called cusp groups, because they can be explained in terms of pinching points on surfaces to cusps. Some groups, like the ones in our pictures here, have one ‘extra’ parabolic element (in this case b) and so are called single cusps.