Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds

Abstract

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group $Δ_{4,4,\infty;\infty}$. By comparing the combinatorial structures of the Ford domain of $Δ_{4,4,\infty;\infty}$ and the Dirichlet domain of $Δ_{4,4,n;\infty}$, we prove that for each $n \geq 5$, the Dehn filling of $s782$ along the slope $(n-1)\mathcal{m}_1 + \mathcal{l}_1$ on its second cusp admits a spherical CR uniformization, where $(\mathcal{m}_1, \mathcal{l}_1)$ denotes the meridian-longitude system of a cusp in SnapPy notation.