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Ideal triangulations of hyperbolic 3-manifolds. (English) Zbl 0983.57013
Proceedings of the 16th national congress of the Italian Mathematical Society, Napoli, Italy, September 13-18, 1999. Bologna: Unione Matematica Italiana, 593-608 (2000).
This paper deals with the problem of geodesic ideal triangulations of oriented, complete, finite volume, non-compact hyperbolic 3-manifolds. The conjecture is that every such hyperbolic 3-manifold admits a geodesic ideal triangulation into (non-degenerate) ideal tetrahedra; this idea forms the basis for the construction of the hyperbolic structure on the manifold, for example in the program SNAPPEA by J. Weeks. One associates a modulus in the upper half plane to each edge of an ideal tetrahedron. From the compatibility condition about each edge and the angle condition, we get a system of equations in the moduli. To get a complete structure, we also require that the induced affine structure on the boundary tori are flat which gives another system of equations for the moduli. The solution of these equations gives a complete hyperbolic structure on the manifold. However, to date, it is only known that partially flat geodesic ideal triangulations exist, by results of Epstein and Penner. In this case, some ideal tetrahedra may be flat, corresponding to some edges having real modulus and the question arises of whether the converse of Epstein and Penner’s result is true, that is, whether a set of such modulus which satisfy the equations stated above gives a complete hyperbolic structure on the manifold. The paper gives an exposition (without proof) of some of the results obtained by the author and J. Porti [Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, to appear in Expos. Math.] and also by the author and J. Weeks [Partially flat ideal triangulations of hyperbolic 3-manifolds, to appear in Osaka J. Math.], mainly concerning the converse of the existence result of Epstein and Penner and also problems concerning the deformation of such partially flat ideal triangulations. He gives in detail an example in dimension 2 of how the topology of the identification space in such a case can degenerate. Several questions and interesting conjectures are also posed in the paper about what conditions are necessary and sufficient for the identification space obtained from the equations to be non-degenerate.
For the entire collection see [Zbl 0953.00026].

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)