Jørgensen's doubly-degenerate group

The example in Figure 10.9 is an approximation to a true sphere-filling limit set discovered by Jorgensen. Jørgensen actually discovered a whole category of groups of Möbius maps whose limit sets were later described by Cannon and Thurston as ‘natural sphere-filling curves’, as if they were suitable for sale in the organic produce section of the grocery. One of the remarkable feature of these groups is that they are still discrete in the sense we discussed on p.239ff., even though their limit sets fill up the whole sphere. When there is no regular set, it is not at all obvious how to tell when a group is discrete, although in fact it is a theorem that any limit of discrete groups is itself discrete. In the case of Jørgensen's groups there are deeper reasons connected with three-dimensional hyperbolic geometry, to be touched on in the final chapter.

One way to understand Jørgensen's doubly-degenerate group is to see it as a limit of double cusp groups, using a form of the sock trick explained in the last section. The idea is that we shall start with the sequence of double cusp groups with cusps 1/0, P/Q which we used to converge to Jørgensen's singly-degenerate group, and use Grandma's party trick explained on p.299 in the last chapter to conjugate these groups into groups where the cusps are of the form p/q, −q/p(where q is quite a bit smaller than Q).