Kirby Problem 5.14

Definitions and mathematical statement

A finite-volume hyperbolic $n$-manifold is a connected complete Riemannian manifold of constant sectional curvature $-1$ whose total Riemannian volume is finite. In the noncompact case, each end is a cusp: outside a compact set it is modeled on $F\times[0,\infty)$ with metric $dt^2+e^{-2t}g_F$, where $(F,g_F)$ is a closed flat $(n-1)$-manifold. Equivalently the manifold is a quotient $\mathbb H^n/\Gamma$ by a torsion-free lattice.

For each integer $n\ge5$, does there exist such a manifold with exactly one cusp? The cusp condition counts ends, not boundary components of an arbitrarily chosen compact subset. Orientability is not required in the stated question; an orientable version would be a further restriction.

The source's phrase "for any $n\ge5$" can also motivate the weaker existence question: is there even one dimension at least five admitting a one-cusped example? Record the dimension-by-dimension question and this weaker test separately, rather than conflating their quantifiers.

The group need not be arithmetic, and no fixed flat cross-section is prescribed. An orbifold with torsion in its lattice does not meet the manifold requirement. A construction with several cusps likewise does not suffice, even if those cusps are isometric or permuted transitively by symmetries. Passing to a finite cover is allowed as a method, but the resulting number of ends must still be exactly one.

Short English statement

In which dimensions at least five do complete finite-volume hyperbolic manifolds with exactly one cusp exist?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3021 (KP-5.14), 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 5.14. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.