Kirby Problem 3.30

Definitions and mathematical statement

Let $M$ be a compact orientable three-manifold specified by a triangulation with $t$ tetrahedra. In the finite-volume convention, $M$ is hyperbolic if its interior admits a complete sectional-curvature $-1$ metric of finite volume; the closed case is included, and nonempty cusp boundary consists of tori. This convention is different from prescribing geodesic boundary.

(a) Determine the bit complexity of deciding whether $M$ is hyperbolic. (b) For a positive instance, determine the complexity of constructing a finite, verifiable description of its hyperbolic structure. (c) Can the decision problem lie in NP and the construction problem in FNP?

For precision, membership in NP requires polynomial-length certificates whose correctness a deterministic algorithm checks in time polynomial in the triangulation input length. The FNP version asks for a polynomially balanced, polynomial-time checkable relation between the input and a finite description of a structure. Such a description must specify how exact geometric parameters are encoded and certified, for example algebraic parameters with isolating data and verified gluing and completeness conditions in a suitable geometric decomposition.

Uncertified floating-point approximations are not such certificates. A solution should state its output representation and prove both geometric validity and the claimed size and verification-time bounds.

Short English statement

How difficult is it to recognize and construct a hyperbolic structure from a three-manifold triangulation, and can valid structures have efficiently checkable certificates?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2828 (KP-3.30), 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 3.30. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.