Kirby Problem 5.13

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.

A finite-volume hyperbolic $n$-manifold is a connected complete Riemannian $n$-manifold of sectional curvature $-1$ with finite total volume; it may be closed or cusped. A finite cover here is a connected covering of finite degree. Its first Betti number is $b_1=\dim_{\mathbb Q}H_1(-;\mathbb Q)$. In the source's usage, a cubulated group is the fundamental group of a compact nonpositively curved cube complex, equivalently it acts freely and cocompactly on that complex's CAT(0) universal cover. Fibering over $S^1$ means a locally trivial manifold bundle over the circle.

Statement recorded in the supplied source.

For every finite-volume hyperbolic $n$-manifold $M$, resolve the following four assertions (the finite cover may differ between parts):

1. (a) Some finite cover has $b_1>0$.
2. (b) Some finite cover has fundamental group admitting a surjection onto a nonabelian free group.
3. (c) Some finite cover has cubulated fundamental group.
4. (d) If $n$ is odd, some finite cover fibers over $S^1$.

[S2]

Short English statement

Do finite-volume hyperbolic manifolds virtually have positive first Betti number, a free-group quotient, a cubulation, and, in odd dimension, a fibration over the circle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3020 (KP-5.13). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 5.13, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.