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

Abstract Let G6,3= ⟨a0, · · · , a5|a3i = id, aiai+1 = ai+1ai, i ∈ Z/6Z⟩ be a hyperbolic group with boundary the Menger curve. Granier [10] constructed a discrete, convex cocompact and faithful representation p of G6,3 into PU(2,1). We show the 3-orbifold at infinity of p(G6,3 )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 [14], and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits uniformizable spherical CR-structure after Schwartz's first example in [25].