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Complex hyperbolic geometry of the figure-eight knot. (English) Zbl 1335.32028
Any real hypersurface \(X\subset {\mathbb C}^2\) inherits from the ambient space a CR-structure (the largest subbundle of the tangent bundle that is invariant under the complex structure). Such a structure is called spherical if it is locally equivalent to the CR structure of \(S^3\), i.e., if there is an atlas on \(X\) of charts with values in \(S^3\) and with transition functions given by restrictions of biholomorphisms of the ball \({\mathbb B}^2\). A spherical CR-structure on \(X\) is called uniformizable if \(X\) can be obtained as a quotient \(X=\Gamma\backslash\Omega\), where \(\Omega\) an open subset of the sphere \(S^3\) and \(\Gamma\subset PU(2,1)\) a discrete subgroup acting properly discontinuously without fixed points on \(\Omega\).
In this paper the CR structure of the complement of the figure-eight knot complement \(M\) is investigated. The manifold \(M\) can be triangulated by just two tetrahedra. In [E. Falbel, J. Differ. Geom. 79, No. 1, 69–110 (2008; Zbl 1148.57025)], all solutions of the compatibilty equations were given for this triangulation to give a spherical CR structure on \(M\). There are only three solutions (up to complex conjugation), yielding three representations \(\rho_i: \Pi_1(M)\to PU(2,1)\), and discrete subgroups \(\Gamma_i= \rho_i(\Pi_1(M))\) of \(PU(2,1)\), for \(i=1,2,3\).
In [Zbl 1148.57025] it was also shown that \(\rho_1\) is the holonomy of a branched spherical CR structure, and that no spherical CR structure with holonomy \(\rho_1\) is uniformizable. In this paper it is shown that \(\rho_2\) and \(\rho_3\) are obtained from each other by an orientation switch, and the following theorem is proved for \(\rho:=\rho_2\) and \(\Gamma:=\Gamma_2\).
Theorem. The domain of discontinuity \(\Omega\) of \(\Gamma\) is nonempty. The action of \(\Gamma\) has no fixed points in \(\Omega\), and the quotient \(\Gamma\backslash\Omega\) is homeomorphic to the figure-eight knot complement.
In other words, the figure-eight knot complement admits a spherical CR uniformization. It is also proved that such a uniformization is unique under the requirement that the boundary holonomy be unipotent.

MSC:
32V05 CR structures, CR operators, and generalizations
57M50 General geometric structures on low-dimensional manifolds
22E40 Discrete subgroups of Lie groups
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