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Triangulations of hyperbolic 3-manifolds admitting strict angle structures. (English) Zbl 1262.57018

It is conjectured that every cusped hyperbolic \(3\)-manifold admits a geometric triangulation, that is, each tetrahedron of the triangulation is a positive volume ideal hyperbolic tetrahedron. A necessary condition of a topological ideal triangulation being geometric is that the triangulation is angled, i.e., admits strict angle structures.
In this interesting paper, the authors construct angled triangulations of cusped hyperbolic \(3\)-manifolds, under the assumption that if the manifold is homeomorphic to the interior of a compact \(3\)-manifold \(\overline M\) with torus or Klein bottle boundary components, then \(H_1(\overline M;\mathbb Z_2)\rightarrow H_1(\overline M,\partial\overline M;\mathbb Z_2)\) is the zero map. As a consequence, each hyperbolic link complement in \(S^3\) admits angled triangulations.
The triangulations are obtained by carefully subdividing Epstein-Penner’s polyhedral decomposition into tetrahedra and inserting flat tetrahedra. To show that the constructed triangulations are angled, a result of Kang-Rubinstein and Luo-Tillmann relating the existence of strict angle structures and the non-existence of certain vertical normal surface classes is used; and the homological assumption on the manifold rules out the existence of those vertical normal surface classes. The authors also explain that the angled triangulations they construct are in general not geometric.

MSC:

57M50 General geometric structures on low-dimensional manifolds

Software:

SnapPea; Snap
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