zbMATH — the first resource for mathematics

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.

32V05 CR structures, CR operators, and generalizations
57M50 General geometric structures on low-dimensional manifolds
22E40 Discrete subgroups of Lie groups
Full Text: DOI arXiv
[1] A F Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer (1995)
[2] N Bergeron, E Falbel, A Guilloux, Tetrahedra of flags, volume and homology of \(\mathrm{SL}(3)\), Geom. Topol. 18 (2014) 1911 · Zbl 1365.57023
[3] D Burns Jr., S Shnider, Spherical hypersurfaces in complex manifolds, Invent. Math. 33 (1976) 223 · Zbl 0357.32012
[4] M Deraux, Deforming the \(\mathbb R\)-Fuchsian \((4,4,4)\)-triangle group into a lattice, Topology 45 (2006) 989 · Zbl 1120.20052
[5] E Falbel, A spherical \(\mathrm{CR}\)-structure on the complement of the figure eight knot with discrete holonomy, J. Differential Geom. 79 (2008) 69 · Zbl 1148.57025
[6] E Falbel, P V Koseleff, F Rouiller, Representations of fundamental groups of \(3\)-manifolds into \(\mathrm{PGL}(3,\mathbbC)\): Exact computations in low complexity, to appear in Geom. Dedicata · Zbl 1326.57041
[7] E Falbel, J Wang, Branched sperical \(\mathrm{CR}\)-structures on the complement of the figure-eight knot, Michigan Math. J. 63 (2014) 635 · Zbl 1300.32034
[8] S Garoufalidis, M Goerner, C K Zickert, Gluing equations for \(\mathrm{PGL}(n,\mathbbC)\)-representations of \(3\)-manifolds, · Zbl 1347.57014
[9] W M Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for \(3\)-manifolds, Trans. Amer. Math. Soc. 278 (1983) 573 · Zbl 0518.53041
[10] W M Goldman, Complex hyperbolic geometry, Clarendon Press (1999) · Zbl 0939.32024
[11] B Maskit, Kleinian groups, Grundl. Math. Wissen. 287, Springer (1988) · Zbl 0627.30039
[12] G D Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980) 171 · Zbl 0456.22012
[13] J R Parker, Complex hyperbolic Kleinian groups, in preparation · Zbl 0793.53069
[14] M B Phillips, Dirichlet polyhedra for cyclic groups in complex hyperbolic space, Proc. Amer. Math. Soc. 115 (1992) 221 · Zbl 0768.53033
[15] R Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975) 281 · Zbl 0309.55002
[16] D Rolfsen, Knots and links, Math. Lecture Series 7, Publish or Perish (1990) · Zbl 0854.57002
[17] R E Schwartz, Degenerating the complex hyperbolic ideal triangle groups, Acta Math. 186 (2001) 105 · Zbl 0998.53050
[18] R E Schwartz, Real hyperbolic on the outside, complex hyperbolic on the inside, Invent. Math. 151 (2003) 221 · Zbl 1039.32030
[19] R E Schwartz, Spherical \(\mathrm{CR}\)-geometry and Dehn surgery, Annals of Math. Studies 165, Princeton Univ. Press (2007) · Zbl 1116.57016
[20] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.