Kirby Problem 3.17

Definitions and mathematical statement

For a complete finite-volume hyperbolic $n$-manifold $M$, $n\geq2$, its length spectrum is the multiset of lengths of closed geodesics. One may first list primitive unoriented closed geodesics, counting distinct geometric geodesics of the same length separately; including every positive iterate then gives the full spectrum. Use the same convention for both manifolds. Equality of the full spectra is equivalent to equality of the primitive multisets after the contributions of iterates are removed.

Two hyperbolic manifolds are commensurable if there exists a hyperbolic manifold $N$ with finite-sheeted local-isometry covering maps to both. This is weaker than being isometric, since distinct finite covers of one manifold may have identical length spectra.

Suppose $M_1$ and $M_2$ are complete finite-volume hyperbolic manifolds of the same dimension and have equal length spectra with multiplicity. Must they be commensurable? The lengths are real translation lengths of geodesics; no extra normal-bundle holonomy angles are supplied. Agreement up to an arbitrary but finite cutoff is insufficient, and agreement only as sets after discarding multiplicities is a different hypothesis. The assertion ranges over both arithmetic and nonarithmetic manifolds.

Short English statement

If two finite-volume hyperbolic manifolds have exactly the same closed-geodesic lengths with multiplicity, must they have a common finite cover?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2815 (KP-3.17), 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 3.17. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.