#### Volume 20, issue 6 (2016)

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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
##### Abstract

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.

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