Complex Hyperbolic Geometry of Chain Links

The complex hyperbolic triangle group =₄, , ; acting on the complex hyperbolic plane H² ₂ is generated by complex reflections I₁, I₂, I₃ such that the product I₂I₃ has order four, the products I₃I₁, I₁I₂ are parabolic and the product I₁I₃I₂I₃ is an accidental parabolic element. Unexpectedly, the product I₁I₂I₃I₂ is a hidden accidental parabolic element. We show that the 3-manifold at infinity of ₄, , ; is the complement of the chain link 8⁴₁ in the 3-sphere. In particular, the quartic cusped hyperbolic 3-manifold S³-8⁴₁ admits a spherical CR-uniformization. The proof relies on a new technique to show that the ideal boundary of the Ford domain is an infinite-genus handlebody. Motivated by this result and supported by the previous studies of various authors, we conjecture that the chain link Cₚ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group ₏, ₐ, ₑ;, for 3 p 9.