This morning there was a paper which caught my eye:
Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.
In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold.
Geometrization contains a characterization of manifolds that admit a geometry modeled on real hyperbolic 3-space. CR-geometry, on the other hand, is about manifolds which admit a geometry that is locally modeled on the CR structure (the largest subbundle in the tangent bundle that is invariant under the complex structure) of viewed as the boundard of the unit ball . Such a structure is called uniformizable roughly if our manifold M has a discrete cover which sits inside , with the CR-structure on M lifting to the standard CR-structure on (I think).
Quotients of such as lens spaces give the simplest examples of manifolds with uniformizable spherical CR-structures. A nice question, which is a bit reminiscent of a part of Geometrization, is to topologically classify which manifolds admit such structures, and how many of them there are.
CR structures are very good to have around; CR-structures are the kind of structures you get on real hypersurfaces in complex manifolds. There is a whole theory around them which parallels Riemannian geometry, and there seems to be a lot of deep analysis going on (which I know next to nothing about) around trying to understand various fundamental operators and their spectra (sub-Laplacian, Kohn Laplacian…).
This paper gives a really interesting example of a manifold with a uniformizable spherical CR-structure, namely the figure-eight knot complement. This manifold played an important motivational role in the development of real hyperbolic geometry, and the hope is that here too it will provide a good motivational example to study, which is simple enough to work with “by hand” but complicated enough to exhibit somehow “generic” behaviour.
A quite fascinating research programme! Looking at the a class of manifolds as manifolds at infinity of complex hyperbolic orbifolds! I look forward to reading this paper and to learning more about this!