A spherical CR structure on the complement of the figure eight knot with discrete holonomy

Abstract

We describe a general geometrical construction of spherical CR structures. We construct then spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in $PU(2,1,{\bf Z}[\omega])$ and $PU(2,1,{\bf Z}[i])$ respectively. These are the same ring of integers appearing in the real hyperbolic geometry of the corresponding links.