Monsters in disguise

Early in this chapter, we sailed down into the deeper southern reaches of the Maskit boundary by carefully avoiding the jutting rocky promontories where the cusp groups sit. To reach Jørgensen's degenerate group at the furthermost point, we had to head through the slice holding a tight almost due south course without very much room for straying. However the cusp groups, by virtue of their location on promontories, can either be approached straight on heading south from the interior or by laying up and following up northwards along the edge of the rocky promontories which lead to the cusps. The coastline route, not unlike the coast of Maine, has surprises in store.

Let's return to the most prominent cusps of Maskit's slice corresponding to 0/1, at the value μ = 2i. As we have seen, the limit set of this group is none other than the Apollonian gasket. A direct approach from the interior of Maskit's slice would be to sail due south down the imaginary axis, that is, through groups with μ = ti with real numbers t decreasing to 2. We have already shown several pictures of this kind of direct ‘pinching’, where gaps are slowly closed to pinch points (for example, see Figure 7.4). Nothing terribly exciting happens during these trips.