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.