##
**Ideal triangulations of 3-manifolds II; taut and angle structures.**
*(English)*
Zbl 1096.57018

This is the second of a series of five papers in which the authors investigate ideal triangulations of the interiors of compact \(3\)-manifolds. The first paper of the series [E. Kang and J. Hyam Rubinstein, Proceedings of the Casson Fest, Geom. Topol. Monogr. 7, 235–265 (electronic) (2004; Zbl 1085.57016)] examined normal surface theory for ideal triangulations.

Let \(M\) be a compact \(3\)-manifold having nonempty boundary, each component of which is a torus or Klein bottle. An angle structure for an ideal triangulation \(\Gamma\) of \(M\) is an assignment of positive dihedral angles at the edges of the ideal tetrahedra, such that opposite edges of each ideal tetrahedron have the same angles, the total of the six angles for each ideal tetrahedron is \(2\pi\), and the total angle around each edge is also \(2\pi\). The existence of such an assignment is one of Thurston’s three hyperbolic gluing conditions; if all are satisfied, then the hyperbolic structures determined on the ideal tetrahedra fit together to give a complete, finite-volume hyperbolic structure on \(M\). Also, the existence of an angle structure implies that the fundamental group of \(M\) is \(\text{CAT}(0)\), and is relatively word hyperbolic.

Two weaker notions are used by the authors: a semiangle structure requires only that the angles be non-negative, and a taut structure (a slight weakening of tautness as defined and studied by M. Lackenby [Invent. Math. 140, No. 2, 243–282 (2000; Zbl 0947.57016)] is a semiangle structure in which each angle is either \(0\) or \(\pi\). Lackenby showed that ideal triangulations with taut structures always exist when \(M\) is orientable, irreducible, and atoroidal.

A. Casson and I. Rivin proved that if \(\Gamma\) admits an angle structure, then \(M\) is irreducible, \(\mathbf{P}^2\)-irreducible and atoroidal. Moreover, \(\Gamma\) is strongly \(1\)-efficient, a condition on ideal triangulations developed by the second author and W. Jaco. The first main result of the paper is a version of this for semiangle structures. Its conclusion is that \(M\) is irreducible, \(\mathbf{P}^2\)-irreducible, and \(\Gamma\) is \(0\)-efficient, and any imbedded normal torus or Klein bottle is incompressible. If in addition \(M\) is atoroidal, then \(\Gamma\) is strongly \(1\)-efficient.

The second half of the paper investigates the existence of angle structures on a taut ideal triangulation. The main result is that if \(M\) is atoroidal and \(\Gamma\) admits a taut or semiangle structure, then \(\Gamma\) admits an angle structure (actually, a \(k\)-dimensional space of them, where \(k\) is the number of tetrahedra of \(\Gamma\)) if and only if there are no branched normal classes with angle sum \(2\pi\) in the taut or semiangle structure, and no quadrilaterals with angle sum \(<2\pi\). A normal class is a numerical solution to the normal surface compatibility equations of the triangulation; a solution consisting of non-negative integers defines a normal surface which may be immersed and may have branching.

The authors use their main result to give examples of taut ideal triangulations which do not admit angle structures, answering a question of Lackenby in the negative. Also, the case of punctured-torus bundles over the circle is examined in detail.

Let \(M\) be a compact \(3\)-manifold having nonempty boundary, each component of which is a torus or Klein bottle. An angle structure for an ideal triangulation \(\Gamma\) of \(M\) is an assignment of positive dihedral angles at the edges of the ideal tetrahedra, such that opposite edges of each ideal tetrahedron have the same angles, the total of the six angles for each ideal tetrahedron is \(2\pi\), and the total angle around each edge is also \(2\pi\). The existence of such an assignment is one of Thurston’s three hyperbolic gluing conditions; if all are satisfied, then the hyperbolic structures determined on the ideal tetrahedra fit together to give a complete, finite-volume hyperbolic structure on \(M\). Also, the existence of an angle structure implies that the fundamental group of \(M\) is \(\text{CAT}(0)\), and is relatively word hyperbolic.

Two weaker notions are used by the authors: a semiangle structure requires only that the angles be non-negative, and a taut structure (a slight weakening of tautness as defined and studied by M. Lackenby [Invent. Math. 140, No. 2, 243–282 (2000; Zbl 0947.57016)] is a semiangle structure in which each angle is either \(0\) or \(\pi\). Lackenby showed that ideal triangulations with taut structures always exist when \(M\) is orientable, irreducible, and atoroidal.

A. Casson and I. Rivin proved that if \(\Gamma\) admits an angle structure, then \(M\) is irreducible, \(\mathbf{P}^2\)-irreducible and atoroidal. Moreover, \(\Gamma\) is strongly \(1\)-efficient, a condition on ideal triangulations developed by the second author and W. Jaco. The first main result of the paper is a version of this for semiangle structures. Its conclusion is that \(M\) is irreducible, \(\mathbf{P}^2\)-irreducible, and \(\Gamma\) is \(0\)-efficient, and any imbedded normal torus or Klein bottle is incompressible. If in addition \(M\) is atoroidal, then \(\Gamma\) is strongly \(1\)-efficient.

The second half of the paper investigates the existence of angle structures on a taut ideal triangulation. The main result is that if \(M\) is atoroidal and \(\Gamma\) admits a taut or semiangle structure, then \(\Gamma\) admits an angle structure (actually, a \(k\)-dimensional space of them, where \(k\) is the number of tetrahedra of \(\Gamma\)) if and only if there are no branched normal classes with angle sum \(2\pi\) in the taut or semiangle structure, and no quadrilaterals with angle sum \(<2\pi\). A normal class is a numerical solution to the normal surface compatibility equations of the triangulation; a solution consisting of non-negative integers defines a normal surface which may be immersed and may have branching.

The authors use their main result to give examples of taut ideal triangulations which do not admit angle structures, answering a question of Lackenby in the negative. Also, the case of punctured-torus bundles over the circle is examined in detail.

Reviewer: Darryl McCullough (Norman)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57Q15 | Triangulating manifolds |

### Keywords:

triangulation; ideal; normal; angle; semiangle; taut; hyperbolic; structure; gluing; CAT(0); relatively hyperbolic; efficient; Pachner### Software:

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\textit{E. Kang} and \textit{J. H. Rubinstein}, Algebr. Geom. Topol. 5, 1505--1533 (2005; Zbl 1096.57018)

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