Partially flat ideal triangulations of cusped hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Partially flat ideal triangulations of cusped hyperbolic 3-manifolds
zbMATH Open document ID: 0952.57003
author: Carlo Petronio
author: Jeffrey R. Weeks
published in: Osaka Journal of Mathematics
publication date: 2001-01-08T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1515486
zbMATH Keywords: geodesic
zbMATH Keywords: 3-manifold
zbMATH Keywords: triangulation
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1515486