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

Abstract

Let \(G_{6,3}\) be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation \(\rho \) of \(G_{6,3}\) into \(\textbf{PU}(2,1)\). 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 Kapovich’s Conjecture 10.6 in Kapovich (in: In the tradition of thurston II. Geometry and groups, Springer, Cham, 2022), and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz’s first example in Schwartz (Invent Math 151(2):221–295, 2003).