Complex Hyperbolic Geometry of Chain Links

Abstract

The complex hyperbolic triangle group $Γ=Δ_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane ${\bf H}^2_{\mathbb C}$ is generated by complex reflections $I_1$, $I_2$, $I_3$ such that the product $I_2I_3$ has order four, the products $I_3I_1$, $I_1I_2$ are parabolic and the product $I_1I_3I_2I_3$ is an accidental parabolic element. Unexpectedly, the product $I_1I_2I_3I_2$ is a hidden accidental parabolic element. We show that the 3-manifold at infinity of $Δ_{4,\infty,\infty;\infty}$ is the complement of the chain link $8^4_1$ in the 3-sphere. In particular, the quartic cusped hyperbolic 3-manifold $S^3-8^4_1$ 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 $Δ_{p,q,r;\infty}$, for $3 \leq p \leq 9$.

Figures (13)

• Figure 1: The chain links $8^4_1$ (the left) and $10^5_1$ (the right). Link diagrams from https://knotinfo.math.indiana.edu/
• Figure 2: A realistic view the ideal boundary of the Ford domain of $\Delta(4, \infty,\infty;\infty)$.
• Figure 3: An abstract picture of the Ford domain of $\Sigma$. For example, the sphere labeled by $B$ is the spinal sphere $\partial_{\infty}I(B)$, in the notation of Proposition \ref{['prop:pair-disjoint4ppp']}, it is $\partial_{\infty}I^{+}_0$.
• Figure 4: 3-sides of $D_{\Sigma}$. The left one is $s^\star_0$, the two shaded disks are $s^\star_0 \cap \partial {\bf H}^2_{\mathbb C}$. The central one is $s^{+}_0$, the $B^{-1}$-labeled sectors are $s^{+}_0 \cap s^{-}_0$, the $B^{2}$-labeled sectors are $s^{+}_0 \cap s^{\star}_0$. The right one is $s^{-}_0$, the labelled sectors have similar meanings. The cyan point in each sub-figure is the fixed point of $B$.
• Figure 5: The disks $E_{B}$ (the union of the blue and green triangles) and $E_{B^{-1}}$ (the union of the cyan and yellow triangles).

Theorems & Definitions (48)

• Conjecture 1.1: Schwartz schwartz-icm
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4: Goldman Go, Parker and Will ParkerWill:2017
• Proposition 2.5: Goldman Go, Parker and Will ParkerWill:2017
• Remark 2.6
• Definition 2.7
• Proposition 2.8: Go, Section 5.4.5