Kirby Problem 3.35

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a compact connected $3$-manifold $M$, let $r(M)$ be the least cardinality of a generating set of $\pi_1(M)$. A Heegaard splitting writes $M$ as two compression bodies meeting along their common positive boundary; a compression body is obtained from a surface times an interval by adding $1$-handles, with a handlebody allowed when the negative boundary is empty. The Heegaard genus $g(M)$ is the minimum genus of that common surface. A closed orientable irreducible manifold is Haken if it contains a closed embedded two-sided incompressible surface of positive genus; incompressibility means injection of its fundamental group. Write $\nu(K)$ for an open tubular neighborhood of a knot. For an arithmetic hyperbolic manifold $M=\mathbb H^3/\Gamma$, a congruence cover corresponds to a finite-index congruence subgroup of the defining arithmetic lattice $\Gamma$ (a subgroup containing a principal congruence subgroup for some nonzero ideal). Use the complete hyperbolic volume $\operatorname{vol}(M)$.

Statement recorded in the supplied source.

Resolve all five questions:

1. If $M$ is closed, orientable and hyperbolic and $r(M)=2$, must $g(M)=2$?
2. Does a non-Haken $3$-manifold $M$ exist with $r(M)<g(M)$?
3. Is there a constant $k>0$, independent of $M$, such that $g(M)\le k\,r(M)$ for every closed orientable $3$-manifold?
4. For each arithmetic hyperbolic $3$-manifold $M$, is $\inf_i r(M_i)/\operatorname{vol}(M_i)$ zero or positive, where $M_i$ ranges over all its congruence covers?
5. Does some knot $K\subset S^3$ satisfy $r(S^3\setminus\nu(K))<g(S^3\setminus\nu(K))$?

[S2]

Short English statement

Compare the minimum number of fundamental-group generators with Heegaard genus: in rank-two hyperbolic manifolds, non-Haken manifolds, all closed orientable manifolds, arithmetic congruence covers, and knot exteriors.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2833 (KP-3.35). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.35, p. 156 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.