Spherical CR uniformization of the magic 3-manifold

TL;DR

This paper demonstrates that certain complex hyperbolic triangle groups have 3-manifolds at infinity that are known hyperbolic 3-manifolds, supporting a conjecture relating these groups to Dehn fillings of the magic 3-manifold.

Contribution

It establishes explicit connections between complex hyperbolic triangle groups and specific hyperbolic 3-manifolds at infinity, proposing a conjecture on their uniformization via Dehn fillings.

Findings

The 3-manifold at infinity of $ riangle_{3, ext{infinity}, ext{infinity}; ext{infinity}}$ is the magic 3-manifold $6_1^3$.

The 3-manifold at infinity of $ riangle_{3,4, ext{infinity}; ext{infinity}}$ is the manifold $m295$ from the Snappy Census.

These manifolds admit spherical CR uniformizations.

Abstract

We show the 3-manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,\infty,\infty;\infty}$ is the three-cusped "magic" 3-manifold $6_1^3$. We also show the 3-manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,4,\infty;\infty}$ is the two-cusped 3-manifold $m295$ in the Snappy Census, which is a 3-manifold obtained by Dehn filling on one cusp of $6_1^3$. In particular, hyperbolic 3-manifolds $6_1^3$ and $m295$ admit spherical CR uniformizations. These results support our conjecture that the 3-manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,n,m;\infty}$ is the one-cusped hyperbolic 3-manifold from the "magic" $6_1^3$ via Dehn fillings with filling slopes $(n-2)$ and $(m-2)$ on the first two cusps of it.