Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

Abstract

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane H 2 C . We deform the Ford domain of Parker and Will in H 2 C in a one parameter family. On the one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer n ≥ 4, and the spherical CR structure obtained for n = 4 is the Deraux-Falbel spherical CR uniformization of the Figure Eight knot complement.

Geometric structures and configurations of flags in orbits of real forms

“…For n = 2 , G 0 = PGL(2, ℝ) and it is clearly true. For n = 3 , G 0 = PGL(3, ℝ) or PU (2,1). In [23], we proved that for PU(2, 1) the invariant ( ) lies in P(ℂ) + .…”

“…3). In particular, there are now examples of CR structures on complement of knots and links which admit hyperbolic metrics of finite volume and many of their Dehn surgeries (see [1,7,10,15,47]). One imposes that the flat structure on the boundary has parabolic holonomy (see Sect.…”

“…[18,46], we know that for any t ∈ ℂ⧵{0, 1} , [t] + [t −1 ] = 0 in P(ℂ) . Hence, it follows from part (1). □ Definition 4.10 Let T ∈ Gc 4 be a generic tetrahedron with edge coordinates z ij (T) , 1 ≤ i ≠ j ≤ 4 .…”

“…12 For a 3-dimensional manifold M which can be triangulated by finitely many tetrahedra, there are interesting P(ℂ)-valued invariants for representations of 1 (M) in PU (2,1) and PGL(3, ℂ) [2,22]. In order to show that the invariants are welldefined, it is crucial to prove that the invariants are independent of the 2-3 moves of the triangulations of M. Theorem 4.11 is equivalent to this.…”

Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold

“…Let H 2 C be the complex hyperbolic plane, the holomorphic isometry group of H 2 C is PU(2, 1). A spherical CR-structure on a smooth 3-manifold M is a maximal collection of distinguished charts modeled on the boundary ∂H 2 C of H 2 C , where coordinates changes are restrictions of transformations from PU (2,1). In other words, a spherical CR-structure is a (G, X)-structure with G = PU(2, 1) and X = S 3 .…”