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

• Published in: Geometriae dedicata Vol. 218; no. 4
• Main Authors: Ma, Jiming, Xie, Baohua
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2024; Springer Nature B.V
• Subjects: Algebraic Geometry; Convex and Discrete Geometry; Differential Geometry; Hyperbolic Geometry; Mathematics; Mathematics and Statistics; Original Paper; Projective Geometry; Topology; 51M10; Spherical CR uniformization; 57M50; Complex hyperbolic geometry; Hyperbolic groups; 22E40; Hyperbolic 3-orbifolds; 20H10; Menger curve
• ISSN: 0046-5755; 1572-9168
• DOI: 10.1007/s10711-024-00934-y

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).