Spherical CR uniformization of the magic 3-manifold

2021/06/12 by Jiming Ma, Ma, Jiming, Baohua Xie +1
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2106.06668

openalex publication_date 2021/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We show the 3-manifold at infinity of the complex hyperbolic triangle group Δ_{3,∞,∞;∞} is the three-cusped "magic" 3-manifold 6₁³. We also show the 3-manifold at infinity of the complex hyperbolic triangle group Δ_3,4,∞;∞ is the two-cusped 3-manifold m295 in the Snappy Census, which is a 3-manifold obtained by Dehn filling on one cusp of 6₁³. In particular, hyperbolic 3-manifolds 6₁³ and m295 admit spherical CR uniformizations. These results support our conjecture that the 3-manifold at infinity of the complex hyperbolic triangle group Δ_3,n,m;∞ is the one-cusped hyperbolic 3-manifold from the "magic" 6₁³ via Dehn fillings with filling slopes (n-2) and (m-2) on the first two cusps of it.

Related

• Three-manifolds at infinity of complex hyperbolic orbifolds
• Complex Hyperbolic Geometry of Chain Links
• Minimum volume cusped hyperbolic three-manifolds
• Triangulations of the `magic manifold' and families of census knots
• Complex hyperbolic triangle groups