×

Straightening cell decompositions of cusped hyperbolic 3-manifolds. (English) Zbl 0927.57012

Let \(M\) be an oriented cusped hyperbolic 3-manifold and let \(\tau\) be a topological ideal triangulation of \(M\). The author gives necessary and sufficient conditions for \(\tau\) to be isotopic to an ideal geodesic triangulation. The basic idea is that one can extend the developing map \(\text{dev}: \widetilde M\to{\mathbb{H}}^3\) to the vertices of \(\widetilde\tau\) to \(\overline {\mathbb{H}^3}\) and look at the induced geodesic triangulation of \(\overline {\mathbb{H}^3}\). The author also gives a characterization for \(\tau\) to flatted into partially flat triangulations and proves that straightening combinatorially equivalent topological ideal cell decompositions gives the same geodesic decomposition, up to isometry.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57Q15 Triangulating manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite