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Complex hyperbolic geometry of the figure-eight knot

Martin Deraux and Elisha Falbel

Geometry & Topology 19 (2015) 237–293
Abstract

We show that the figure-eight knot complement admits a uniformizable spherical CR structure, ie it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.

Keywords
spherical CR structures, geometric structures on $3$–manifolds, complex hyperbolic geometry
Mathematical Subject Classification 2010
Primary: 32V05, 57M50
Secondary: 22E40
References
Publication
Received: 27 March 2013
Revised: 11 February 2014
Accepted: 1 May 2014
Published: 27 February 2015
Proposed: Walter Neumann
Seconded: Benson Farb, Danny Calegari
Authors
Martin Deraux
Institut Fourier
Université de Grenoble 1
BP 74
Saint Martin d’Hères, Cedex
France
Elisha Falbel
Institut de Mathématiques
Université Pierre et Marie Curie
4 place Jussieu
F-75252 Paris
France