The complex hyperbolic triangle group Γ=Δ4,∞,∞;∞ acting on the complex hyperbolic plane H²C 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 Δ4,∞,∞;∞ 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 Δp,q,r;∞, for 3 ≤ p ≤ 9.
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Ma et al. (2024) studied this question.