Partially flat ideal triangulations of cusped hyperbolic 3-manifolds
If the interior of a compact 3-manifold has a complete hyperbolic structure of finite volume then it has a geodesic ideal triangulation, in which some of the tetrahedra may degenerate to flat quadrilaterals (with distinct vertices) [\textit{D. B. A. Epstein} and \textit{R. C. Penner}, J. Differ. Geom. 27, No. 1, 67-80 (1988; Zbl 0611.53036)]. This paper provides a converse; if \(M\) is a compact 3-manifold bounded by tori and Klein bottles and there is a family \(\widetilde\tau\) of ideal tetrahedra in \(\mathbb{H}^3\) (some, but not all, possibly degenerate) satisfying the compatibility and completeness equations determined by the combinatorics of a topological ideal triangulation \(\tau\) then \(M\) has a complete hyperbolic structure. Curiously the structure cannot in general be derived directly from \(\widetilde\tau\). (It is not known whether a genuine ideal triangulation with no degenerate tetrahedra must always exist).
scientific article; zbMATH DE number 1515486