Kirby Problem 2.16

Definitions and mathematical statement

For a complete finite-area hyperbolic surface $X$ without boundary, each essential nonperipheral free homotopy class of simple closed curves has a unique closed geodesic representative. Its unmarked simple length spectrum is the multiset \[ \mathcal L_{\rm simp}(X)= \{\!\{\ell_X(\alpha):\alpha\text{ an unoriented simple closed geodesic class}\}\!\}. \] Count one entry for each unoriented class; distinct classes of equal length produce multiplicities. Iterated traversals of a primitive geodesic are not additional simple curves. No labels identifying the classes on a reference surface are retained.

If two hyperbolic surfaces $X,Y$ have equal unmarked simple length spectra, must they be isometric? In particular, ask this for closed orientable surfaces. An isometry may preserve or reverse orientation, since lengths alone contain no orientation choice. Equality means matching the multiplicity of every real length, rather than merely equality of the sets of values. The challenge is to recover the metric from the unlabeled data; the hypothesis does not supply a homeomorphism that matches individual curve classes. Such a supplied marking would give a different, stronger hypothesis.

Short English statement

Does the multiset of lengths of simple closed geodesics determine a hyperbolic surface up to isometry?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2764 (KP-2.16), 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 2.16. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.