Kirby Problem 3.3

Definitions and mathematical statement

Let $M$ be a connected complete noncompact hyperbolic three-manifold of finite volume. A topological ideal triangulation expresses $M$ as finitely many tetrahedra with their vertices removed, whose triangular faces are identified in pairs. The removed vertices correspond to ends of the manifold; repeated identifications within the quotient are allowed.

A geometric ideal tetrahedron in $\mathbb H^3$ is the convex hull of four distinct points on the sphere at infinity $\partial_\infty\mathbb H^3=S^2$, not all lying on one circle. This last condition rules out flat tetrahedra. A geometric ideal triangulation requires the tetrahedra to have positive volume and their face identifications to be hyperbolic isometries, with the resulting metric equal to the given complete metric on $M$.

Does every such $M$ admit a geometric ideal triangulation? The tetrahedron shapes and the combinatorial triangulation may both be chosen. In the orientable case all tetrahedra must have positive orientation relative to the orientation of $M$. A decomposition using flat or negatively oriented tetrahedra does not satisfy this requirement. Neither does a geometric decomposition into more general ideal polyhedra, unless it can be subdivided into tetrahedra compatibly across every glued face.

Short English statement

Can every cusped hyperbolic three-manifold be assembled from genuine positive-volume ideal tetrahedra with its complete hyperbolic metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2801 (KP-3.3), 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 3.3. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.