Kirby Problem 1.88

Definitions and mathematical statement

Let $d$ be a positive squarefree integer and let $\mathcal O_d$ be the ring of integers of the imaginary quadratic field $\mathbb Q(\sqrt{-d})$. Its Bianchi group $\operatorname{PSL}_2(\mathcal O_d)$ acts by orientation-preserving isometries on hyperbolic three-space. For a nonzero ideal $I\subset\mathcal O_d$, define the principal congruence subgroup \[ \Gamma(I)=\ker\left(\operatorname{PSL}_2(\mathcal O_d) \longrightarrow\operatorname{PSL}_2(\mathcal O_d/I)\right). \] A subgroup $\Gamma\le\operatorname{PSL}_2(\mathcal O_d)$ is congruence when it contains some $\Gamma(I)$; equality is not required.

A congruence arithmetic link complement is a manifold $S^3\setminus L$ homeomorphic to $\mathbb H^3/\Gamma$ for such a torsion-free congruence subgroup, allowing conjugation of the lattice in $\operatorname{PSL}_2(\mathbb C)$. Here $L$ is a finite smooth link and the quotient has its complete finite-volume metric. Ask whether there are infinitely many such link complements, up to homeomorphism, as $d$, $I$, and $\Gamma$ vary.

This is the geometric meaning of infinitude of congruence arithmetic links in the source: it counts distinct complements, rather than repeated diagrams or different choices of congruence data for one manifold. A principal congruence example belongs to the class, but restricting to principal subgroups would change the problem. Likewise, commensurability with a Bianchi group without the congruence containment is insufficient.

Short English statement

Are there infinitely many distinct link complements in the sphere whose hyperbolic groups are congruence subgroups of Bianchi groups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2747 (KP-1.88), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.88. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.