This is the second in a series of papers in which we investigate ideal triangulations of
the interiors of compact 3–manifolds with tori or Klein bottle boundaries. Such
triangulations have been used with great effect, following the pioneering work
of Thurston. Ideal triangulations are the basis of the computer program
SNAPPEA of Weeks, and the program SNAP of Coulson, Goodman, Hodgson and
Neumann. Casson has also written a program to find hyperbolic structures on
such 3–manifolds, by solving Thurston’s hyperbolic gluing equations for
ideal triangulations. In this second paper, we study the question of when
a taut ideal triangulation of an irreducible atoroidal 3–manifold admits a
family of angle structures. We find a combinatorial obstruction, which gives a
necessary and sufficient condition for the existence of angle structures for taut
triangulations. The hope is that this result can be further developed to give a
proof of the existence of ideal triangulations admitting (complete) hyperbolic
metrics. Our main result answers a question of Lackenby. We give simple
examples of taut ideal triangulations which do not admit an angle structure.
Also we show that for ‘layered’ ideal triangulations of once-punctured torus
bundles over the circle, that if the manodromy is pseudo Anosov, then the
triangulation admits angle structures if and only if there are no edges of degree 2.
Layered triangulations are generalizations of Thurston’s famous triangulation of
the Figure–8 knot space. Note that existence of an angle structure easily
implies that the 3–manifold has a CAT(0) or relatively word hyperbolic
fundamental
Keywords
normal surfaces, 3–manifolds, ideal triangulations, taut,
angle structures
