Year: 2026
Type: article
Abstract: We study the geometry of the complex hyperbolic triangle group normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity Δ 4 , ∞ , ∞ ; ∞ $\Delta_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane bold upper H Subscript double struck upper C Superscript 2 H C 2 $\mathbf{H}^2_{\mathbb C}$ and show that the 3-manifold at infinity of normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity Δ 4 , ∞ , ∞ ; ∞ $\Delta_{4,\infty,\infty;\infty}$ is the complement of the chain link 8 Subscript 1 Superscript 4 8 1 4 $8^4_1$ in the 3-sphere. In particular, the four-cusped hyperbolic 3-manifold upper S cubed minus 8 Subscript 1 Superscript 4 S 3 − 8 1 4 $S^3-8^4_1$ admits a spherical CR-uniformisation. Since the ideal boundary of the Ford domain is an infinite-genus handlebody, we introduce a new technique to construct embedded topological disks that cap off the genus, ensuring their boundaries are compatible with the side-pairings. This surgical approach facilitates the determination of the fundamental group of the manifold at infinity and simplifies the computation compared to [ 14 ]. Based on this result and previous studies, we conjecture that the chain link upper C Subscript p C p $C_p$ serves as an ancestor of the 3-manifold at infinity for the critical complex hyperbolic triangle group
Authors: JIMING MA, BAOHUA XIE, MENGMENG XU
Institutions: Fudan University, Hunan University, Huaqiao University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1206532262, G4000309048
OpenAlex ID: W4392487987