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

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 ρ of G 6 , 3 into PU (2 , 1) . We show the 3-orbifold at infinity of ρ (G 6 , 3) is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the 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). [ABSTRACT FROM AUTHOR]