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

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
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