11 - Crossing boundaries pages 353 to 365

Summary

He also manifested hundreds of trillions of quadrillions of inconceivable numbers of subtle adornments, which could never be fully described even in a hundred trillion quadrillion inconceivable number of eons…

Avatamsaka Sutra

Despite our many adventures, there remain certain boundaries we have not yet ventured to cross. Across the peaky Maskit boundary is indeed a sea of chaos; but it sparkles with islands of mystery. Here, for many experts, lie the only interesting groups. Another boundary is imposed by our rather artificial restriction to groups with only two generators a and b. Not having further eons at our disposal, all we can do in this short chapter is give a brief glimpse of these further vistas, taking, as Maskit has it, ‘a trip to the zoo’.

Kleinian groups acquired their name from Poincaré. We shall tell more about this story in our epilogue. For our purposes, a Kleinian group will be any discrete group of Möbius transformations. After seeing the plane-filling degenerate limit sets in the last chapter, you will appreciate the delicacy involved when we slip in that little word ‘discrete’.

Closer relations between generators

We begin with Kleinian groups with only two generators. Taking a deep breath, let's venture out to some of those beckoning islands. Figure 11.1 shows what happens if we pick the values t_a ≐ 1.924781−0.047529i, t_b = 2 and t_abAB = 0 and use Grandma's four-alarm special recipe in Box 23.