Rank vs. Genus

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 orientable $3$-manifold $M$, let $r(M)$ be the minimum size of a generating set of $\pi_1(M)$. A Heegaard splitting writes $M$ as two compression bodies meeting along their positive boundary surface $\Sigma$; a compression body is obtained from $F\times[0,1]$ by adding $1$-handles on $F\times\{1\}$, with handlebodies allowed when the negative boundary is empty. Let $g(M)$ be the least genus of such a $\Sigma$. For closed $M$, both pieces are handlebodies. A hyperbolic manifold has a complete finite-volume metric of sectional curvature $-1$ on its interior; closed hyperbolic manifolds are included.

Statement recorded in the supplied source.

Does there exist a hyperbolic $3$-manifold $M$ with $r(M)<g(M)$? How large can the additive difference $g(M)-r(M)$ be, and can it be arbitrarily large? [S2]

Short English statement

Compare the least number of fundamental-group generators with the least Heegaard genus, asking whether strict inequality and unbounded gaps occur in hyperbolic three-manifolds. Li proves both for closed orientable examples. [S3]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3409 (OPG-1783). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Rank vs. Genus, OPG-1783. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/rank_vs_genus.
• S3 Tao Li, Rank and genus of 3-manifolds, Journal of the American Mathematical Society 26 (2013), 777--829; arXiv:1106.6302v2. Introduction, Theorems 1.1 and 1.2. https://arxiv.org/abs/1106.6302v2.