#### Algebraic and Geometric Topology 5 (2005),
paper no. 60, pages 1505-1533.

## Ideal triangulations of 3-manifolds II; taut and angle structures

### Ensil Kang, J Hyam Rubinstein

**Abstract**.
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 group.
**Keywords**.
Normal surfaces, 3-manifolds, ideal triangulations, taut and angle structures

**AMS subject classification**.
Primary: 57M25.
Secondary: 57N10.

**DOI:** 10.2140/agt.2005.5.1505

Submitted: 19 May 2005.
Accepted: 13 June 2005.
Published: 3 November 2005.

Notes on file formats
Ensil Kang, J Hyam Rubinstein

Department of Mathematics, College of Natural Sciences, Chosun University

Gwangju 501--759, Korea

and

Department of Mathematics and Statistics, The University of Melbourne

Parkville, Victoria 3010, Australia

Email: ekang@chosun.ac.kr, rubin@ms.unimelb.edu.au

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