Geometric triangulation conjecture for cusped hyperbolic 3-manifolds

Establish whether every cusped finite-volume hyperbolic 3-manifold admits an ideal triangulation that is geometric, meaning all tetrahedra are positively oriented and convex with respect to the hyperbolic metric.

Background

The authors situate their work in the broader landscape of triangulations in low-dimensional topology. While canonical decompositions into ideal polyhedra exist for cusped hyperbolic 3-manifolds, connecting these to triangulations with desirable geometric properties remains challenging.

They explicitly recall a longstanding conjecture asserting the existence of geometric triangulations for all cusped hyperbolic 3-manifolds, highlighting its status as an open problem.

References

It is an open conjecture that a cusped hyperbolic 3-manifold has a triangulation that is convex with respect to the hyperbolic metric, that is, geometric; see.

— On Geometric triangulations of double twist knots  (2504.09901 - Ibarra et al., 14 Apr 2025) in Introduction, first paragraph

It was conjectured by Thurston that every hyperbolic $3$-manifold with cusp ends has a geometric triangulation. Although still open, this conjecture has been verified for all examples in the SnapPy census.

— $\mathrm {U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$ Turaev-Viro invariants for cusped $3$-manifolds  (2608.16560 - Liu et al., 17 Aug 2026) in Section 1, subsection “U_{q\tilde q}\mathfrak{sl}(2;\mathbb R) Turaev-Viro invariants for hyperbolic 3-manifolds with cusp ends,” immediately before Theorem 1.3 (Theorem \ref{vol})

For complete finite-volume cusped hyperbolic $3$-manifolds, Thurston's geometric triangulation conjecture predicts a decomposition into positive-volume ideal hyperbolic tetrahedra; see .

— Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow  (2609.34108 - Uemura, 28 Sep 2026) in Introduction