Kapovich's unknotted Menger-curve and hyperbolic-quotient conjecture

From papers

Let G6,3=a0,,a5ai3=id, aiai+1=ai+1ai, iZ/6ZG_{6,3}=\langle a_0,\cdots,a_5\mid a_i^3=id,\ a_ia_{i+1}=a_{i+1}a_i,\ i\in\mathbb{Z}/6\mathbb{Z}\rangle be the hyperbolic group with boundary the Menger curve, and let ρ:G6,3PU(2,1)\rho:G_{6,3}\to\mathbf{PU}(2,1) be Granier's discrete, convex-cocompact, faithful representation. Write Λ\Lambda for the limit set of ρ(G6,3)\rho(G_{6,3}), let M\mathcal{M} be the standard Menger curve in R3S3\mathbb{R}^3\subset\mathbb{S}^3, and let Ω\Omega be the domain of discontinuity. Kapovich's conjecture. The limit set Λ\Lambda is ambient-isotopic to M\mathcal{M} in S3\mathbb{S}^3, and the quotient 3-manifold Ω/ρ(G6,3)\Omega/\rho(G_{6,3}) is hyperbolic. The paper's abstract states that it proves the hyperbolicity assertion in the corresponding orbifold setting, while the supplied text does not state a resolution of the unknottedness assertion.

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Sources & referencesView supporting material

Primary source

Jiming Ma and Baohua Xie, “Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold”, arXiv:2201.04765 (2024).

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