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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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