Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold

Abstract

Let $$G_{6,3}=⟨a_0,\cdots ,a_5|a_i^3=id,a_i{a}_{i+1}=a_{i+1}{a}_i,i\in \mathbb{Z}/6\mathbb{Z}⟩$$ be a hyperbolic group with boundary the Menger curve. J. Granier \cite{Granier} constructed a discrete, convex cocompact and faithful representation $\rho$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $\rho (G_{6,3})$ is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the ${\mathbb{Z}}_3$-coned chain-link $C(6,-2)$. This answers the second part of Misha Kapovich's Conjecture 10.6\cite{Kapovich}.

28 pages. arXiv admin note: text overlap with arXiv:1401.0308 by other authors