Low Dimensional Topology

March 9, 2015

Complex hyperbolic geometry of knot complements

Filed under: 3-manifolds,Hyperbolic geometry,Misc. — dmoskovich @ 3:41 am

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 S^3 viewed as the boundard of the unit ball B^2 \subset \mathbb{C}^2. Such a structure is called uniformizable roughly if our manifold M has a discrete cover which sits inside S^3, with the CR-structure on M lifting to the standard CR-structure on S^3 (I think).

Quotients of S^3 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!

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