Slicing it to ribbons

In the mathematical world, one occasionally encounters some rare individuals who seem to have the ability to hold an image in their minds of a four or higher-dimensional object. Even the ability to visualise three-dimensional geometry is a reasonably uncommon gift, and most of us have to get by with two (and sometimes much less) dimensional images in our heads.

The space of quasifuchsian once-punctured torus groups is described by two complex or four real parameters (the traces), which means it is one of these high-dimensional objects. Our approach to studying it has been to look at two-dimensional ‘slices’, meaning that we specify one of the complex trace parameters and then plot those values of the remaining parameter which correspond to quasifuchsian groups (or single cusps, in the case of the Maskit slice).

The samples we have given, Maskit's slice and the trace 3 slice, offer an impression of a object with a somewhat pointy boundary, which is however not terribly complicated otherwise. In recent years, as more detailed plots have emerged, this simple picture has begun to change. Exactly how the slices all fit together is a puzzle of very current interest.