Complex Hyperbolic Geometry of Chain Links

Abstract

The complex hyperbolic triangle group $\Gamma ={\Delta }_{4,\infty ,\infty ;\infty }$ acting on the complex hyperbolic plane ${\mathbf{H}}_{\mathbb{C}}^2$ is generated by complex reflections $I_1$, $I_2$, $I_3$ such that the product $I_2{I}_3$ has order four, the products $I_3{I}_1$, $I_1{I}_2$ are parabolic and the product $I_1{I}_3{I}_2{I}_3$ is an accidental parabolic element. Unexpectedly, the product $I_1{I}_2{I}_3{I}_2$ is a hidden accidental parabolic element. We show that the 3-manifold at infinity of ${\Delta }_{4,\infty ,\infty ;\infty }$ is the complement of the chain link $8_1^4$ in the 3-sphere. In particular, the quartic cusped hyperbolic 3-manifold $S^3-8_1^4$ 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 $C_p$ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group ${\Delta }_{p,q,r;\infty }$, for $3\le p\le 9$.

30 pages, 13 figures