[Paper Review] Geodesic triangulations exist for cusped Platonic manifolds

Matthias Goerner|arXiv (Cornell University)|May 11, 2017

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper proves that every cusped hyperbolic 3-manifold decomposed into isometric Platonic solids—such as dodecahedra or cubes—admits a decomposition into ideal, non-flat geodesic tetrahedra. The construction uses vertex-based diagonal choices on faces to ensure compatibility across gluings, resolving a long-standing question for this class of manifolds.

ABSTRACT

We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra.

Motivation & Objective

• To resolve the open question of whether every cusped hyperbolic 3-manifold admits an ideal geodesic triangulation with non-flat tetrahedra.
• To establish that manifolds decomposed into isometric Platonic solids (e.g., dodecahedra, cubes) can be geometrically triangulated.
• To ensure compatibility of face diagonal choices across cell gluings to maintain a consistent global triangulation.
• To extend prior results that required flat tetrahedra or finite covers by proving existence for the full class of cusped Platonic manifolds.
• To provide a constructive method for building ideal geometric triangulations from canonical cell decompositions.

Proposed method

• Use a lemma stating that a convex ideal polyhedron can be subdivided into non-flat ideal tetrahedra if face diagonals meet at a common vertex.
• Define face cycles in dodecahedral decompositions to organize the global structure of face gluings.
• Assign ownership of faces to dodecahedra such that each dodecahedron owns exactly one face from each opposite pair.
• Apply a five-step diagonal selection process: (1) prioritize dodecahedra with three mutually adjacent faces sharing a vertex; (2) assign diagonals to D-faces randomly; (3) restrict B-face choices based on A-face gluing; (4) resolve dependency cycles by fixing vertex choices on ambiguous edges; (5) recursively assign diagonals to remaining B-faces.
• Ensure that all diagonal choices are compatible across face gluings by enforcing vertex-based meeting conditions and using combinatorial symmetry to classify face patterns.

Experimental results

Research questions

• RQ1 Can every cusped hyperbolic 3-manifold decomposed into isometric Platonic solids be triangulated using only ideal, non-flat geodesic tetrahedra?
• RQ2 Is it possible to subdivide a Platonic cell decomposition into geodesic tetrahedra while preserving face gluing compatibility?
• RQ3 What combinatorial conditions on face diagonals ensure a consistent global triangulation across cell boundaries?
• RQ4 How can dependency cycles in diagonal assignment be resolved without breaking consistency across adjacent cells?
• RQ5 Does the existence of a vertex where all diagonals on adjacent faces meet guarantee a valid geodesic subdivision?

Key findings

• Every cusped hyperbolic Platonic manifold—defined as one decomposable into isometric Platonic solids—admits an ideal geometric triangulation by non-flat geodesic tetrahedra.
• The triangulation is constructed by choosing face diagonals that meet at a common vertex on each face, ensuring compatibility across cell gluings.
• For dodecahedral decompositions, the method resolves dependency cycles by fixing vertex choices on ambiguous edges shared by adjacent B-faces.
• The proof establishes that such triangulations exist even when the dodecahedra are not regular, relying only on isometric decomposition and combinatorial structure.
• The result confirms the existence of ideal geodesic triangulations for the class of cusped Platonic manifolds, extending prior results that required flat tetrahedra or finite covers.
• The construction is algorithmic and combinatorial, relying on face ownership, face cycle orientation, and vertex-based diagonal control to ensure global consistency.