Geometric ideal triangulation conjecture
For every finite-volume cusped hyperbolic -manifold , there exists a finite ideal triangulation of whose tetrahedra are realized as positively oriented geodesic ideal tetrahedra in , with the induced geometric structure equal to the complete hyperbolic structure on .
References
Primary source
Additional references
- Geometric ideal triangulations of hyperbolic 3-manifolds — arXiv — Huabin Ge
Progress summary
A new unrefereed preprint claims that every cusped hyperbolic three-dimensional space has a geometric ideal triangulation, but the claim has not been independently checked.
The conjecture asserts that every finite-volume cusped hyperbolic -manifold admits a geometric ideal triangulation. Earlier literature treated this universal existence statement as open; no proposer or original date is identified in the retrieved sources.
Known results
- Feng and Ge: combinatorial Ricci flow converges exactly when the relevant triangulation is geometric, but universal existence remains open.
- General restricted classes include punctured-torus bundles, two-bridge link complements, certain arborescent link complements, and covers of these spaces.
- The conjecture is true virtually: a finite cover admits a geometric ideal triangulation.
- Zhao, February 5, 2026: an edge-valence bound of at least gives convergence and a geometric truncated-tetrahedral decomposition, improving the previous bound from to .
September 2026 claimed universal proof
On September 23, 2026, a report identified Huabin Ge’s preprint Geometric ideal triangulations of hyperbolic 3-manifolds, which asserts the universal existence statement. If correct, it would remove the longstanding obstruction; the claim is unrefereed and unverified.
Current status (as of September 2026): Ge’s preprint claims a universal solution, but that claim remains unverified; restricted and virtual cases are established, and independent confirmation is needed.
Solutions 0
No solutions have been posted yet.