## 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.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57Q15 Triangulating manifolds

### Citations:

Zbl 1085.57016; Zbl 0947.57016

SnapPea
Full Text:

### References:

 [1] I Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000) 431 · Zbl 0959.57009 [2] B Burton, E Kang, J H Rubinstein, Ideal triangulations of 3-manifolds III: taut structures in low census manifolds, in preparation [3] P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic 3-manifolds, Math. Comp. 68 (1999) 321 · Zbl 0910.57006 [4] D Coulson, O A Goodman, C D Hodgson, W D Neumann, Computing arithmetic invariants of 3-manifolds, Experiment. Math. 9 (2000) 127 · Zbl 1002.57044 [5] M Freedman, J Hass, P Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983) 609 · Zbl 0482.53045 [6] W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245 · Zbl 0100.19402 [7] W Haken, Some results on surfaces in 3-manifolds, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.) (1968) 39 · Zbl 0194.24902 [8] J Hempel, 3-Manifolds, Ann. of Math. Studies 86, Princeton University Press (1976) · Zbl 0345.57001 [9] W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society (1980) · Zbl 0433.57001 [10] W Jaco, J H Rubinstein, 0-efficient triangulations of 3-manifolds, J. Differential Geom. 65 (2003) 61 · Zbl 1068.57023 [11] W Jaco, J H Rubinstein, 1-efficient triangulations of 3-manifolds, in preparation · Zbl 1068.57023 [12] E Kang, Normal surfaces in knot complements, PhD thesis, University of Connecticut (1999) [13] E Kang, Normal surfaces in non-compact 3-manifolds, J. Aust. Math. Soc. 78 (2005) 305 · Zbl 1077.57016 [14] E Kang, J H Rubinstein, Ideal triangulations of 3-manifolds I: Spun normal surface theory, Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 235 · Zbl 1085.57016 [15] M Lackenby, Taut ideal triangulations of 3-manifolds, Geom. Topol. 4 (2000) 369 · Zbl 0958.57019 [16] M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243 · Zbl 0947.57016 [17] E E Moise, Affine structures in 3-manifolds V: The triangulation theorem and Hauptvermutung, Ann. of Math. $$(2)$$ 56 (1952) 96 · Zbl 0048.17102 [18] J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1 · Zbl 0889.57021 [19] J Simon, Compactification of covering spaces of compact 3-manifolds, Michigan Math. J. 23 (1976) · Zbl 0331.57002 [20] M Stocking, Almost normal surfaces in 3-manifolds, Trans. Amer. Math. Soc. 352 (2000) 171 · Zbl 0933.57016 [21] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001 [22] S Tillmann, Degenerations and normal surface theory (2002) [23] J L Tollefson, Normal surface $$Q$$-theory, Pacific J. Math. 183 (1998) 359 · Zbl 0930.57017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.