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A spherical CR structure on the complement of the figure eight knot with discrete holonomy. (English) Zbl 1148.57025

Let \(\Gamma\) be the fundamental group of a 3-manifold and let \(PU(2,1)\) be the group of the homogeneous model for spherical CR-geometry, that is \(\mathbb S^3 \in \mathbb C^2\) with the natural CR-structure induced from the complex structure of \(\mathbb C^2\).
The present paper proposes a geometrical construction of representations of \(\Gamma\) into \(PU(2,1)\), by gluing appropriate tetrahedra adapted to CR-geometry.
In particular, the author constructs (branched) spherical CR-structures on the complement of the figure eight knot and the Whitehead link; these structures are further proved to have discrete holonomies contained in \(PU(2,1, \mathbb Z[\omega])\) and \(PU(2,1, \mathbb Z[i])\), respectively.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
32V05 CR structures, CR operators, and generalizations
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