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A $1$–parameter family of spherical CR uniformizations of the figure eight knot complement

Martin Deraux

Geometry & Topology 20 (2016) 3571–3621

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1–parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1–parameter family of pairwise nonconjugate spherical CR uniformizations of the figure eight knot complement.

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geometric structures, spherical CR structures, complex hyperbolic geometry, discrete groups
Mathematical Subject Classification 2010
Primary: 22E40, 32V05, 57M50
Received: 9 September 2015
Revised: 15 February 2016
Accepted: 18 March 2016
Published: 21 December 2016
Proposed: Walter Neumann
Seconded: Jesper Grodal, Dmitri Burago
Martin Deraux
Institut Fourier
Université Grenoble Alpes
100 rue des maths
38610 Gières