The Menger curve and spherical CR uniformization of a closed hyperbolic \(3\)-orbifold (Q6584166)

A spherical CR-structure on a smooth 3-manifold \(M\) is a maximal collection of distinguished charts modeled on the boundary \(\partial\mathbf{H}^2_{\mathbb{C}}\) of \(\mathbf{H}^2_{\mathbb{C}}\), where coordinate changes are restrictions of transformations from \(\textbf{PU}(2, 1)\), it is said, a spherical CR-structure is a \((G, X)\)-structure with \(G = \textbf{PU}(2, 1)\) and \(X = \mathbb{S}^3\).\N\NGiven two natural numbers \(p\geq5\) and \(q\geq 3\), the polygon-group \(G_{p,q}\) is defined as\N\[\NG_{p,q}=\langle a_0,a_1,\dots,a_{p-1}\ |\ a^q_i=[a_i,a_{i+1}]=id, i \in \mathbb{Z}/p\mathbb{Z}\rangle.\N\]\N\textit{J. Granier} in her doctoral thesis (Université de Fribourg, 2015) constructed a convex-compact representation \(\rho\) of the polygon \(G_{6,3}\) in \(\mathbf{PU}(2,1)\) and described a Dirichlet domain \(D\) for \(\rho\left(G_{6,3}\right)\). She also proved that \(\rho(G_{6,3})\) is discrete. Let \(\Omega\) be the set of discontinuity of the discrete subgroup \(\rho(G_{6,3})\) acting on \(\partial\mathbf{H}^2_{\mathbb{C}}=\mathbb{S}^3\). In this paper, the authors study the topology and geometry of the \(3\)-orbifold \(\Omega / \rho\left(G_{6,3}\right)\) at infinity of \(\rho\left(G_{6,3}\right)\). They prove in Theorem 1.2 that the 3-orbifold \(\Omega / \rho\left(G_{6,3}\right)\) at infinity of \(\rho\left(G_{6,3}\right)\) is a closed hyperbolic 3-orbifold \(\mathcal{O}\), with underlying space the 3-sphere and singularity locus the \(\mathbb{Z}_3\)-coned chain-link \(C(6,-2)\). This result confirms a conjecture posed by \textit{M. Kapovich} and it also is the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure.\N\NThe authors study thoroughly the topology of \(\partial_{\infty} D \cap \Omega\) in \(\mathbb{S}^3\), where \(D\) is a Dirichlet domain centered at the fixed point of an elliptic element of order \(6\) in \(\rho(G_{6,3})\) described by Granier in the above work. They prove in Theorem 1.3 that \(\partial_{\infty} D \cap \Omega\) is a solid torus in the \(3\)-sphere \(\partial \mathbf{H}_{\mathbb{C}}^2\).