SnapPea

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 ef- fect, following the pioneering work of Thurston (22). Ideal triangulations are the basis of the computer program SNAPPEA of Weeks (3) and the program SNAP of Coulson, Goodman, Hodgson and Neumann (4). 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 exam- ples 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. AMS Classification 57M25; 57N10

Abstract: We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure $C$. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of proof of Andreev's Theorem as well as a convenient way to construct three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer sub-multiples of $\pi$, so that the group $\Gamma$ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient $\mathbb{H}^3/\Gamma$ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea. One consequence is a volume estimate for a ``hyperelliptic'' manifold considered by Mednykh and Vesnin (see references).

Abstract: The computer program SnapPea by J. Weeks is a powerful tool for calculating the volumes of hyperbolic 3-manifolds. The small volume hyperbolic 3-manifolds have been studied rather intensively. The smallest known manifold M1, of volume 0.942707, was found independently by A. T. Fomenko and S. V. Matveev, and by J. Weeks. The ten smallest volume manifolds were described in [1, 2] where they were obtained in particular by Dehn surgeries on small volume hyperbolic knots and links. The structure of the set of volumes of hyperbolic 3-orbifolds is given in [3]. The computation of volumes of hyperbolic 3-orbifolds with singular sets obtainable by generalized surgeries on links in the three-dimensional sphere is possible due to SnapPea as well. If the singular set of a 3-orbifold is a graph other than a link, such a general tool is unavailable and the computation of volumes becomes a difficult problem that needs an individual approach in each particular case. It seems thus natural to study 3-orbifolds obtainable by surgeries on hyperbolic 3-orbifolds. The aim of this paper is to study closed hyperbolic 3-orbifolds obtainable by surgery on the smallest cusped hyperbolic 3-orbifolds and study coverings of these orbifolds by hyperbolic 3-manifolds obtainable by surgery on links. Recall [3, 4] that every closed hyperbolic 3-orbifold is obtainable by surgery on an orbifold with a nonrigid cusp (i.e., a cusp on which Dehn surgery, or Dehn filling, can be performed). Three smallest volume hyperbolic 3-orbifolds with nonrigid cusps were described by Adams [4]. Minimal regular coverings of these orbifolds are complements to the well-known links in the three-dimensional sphere. For example, the smallest orbifold with a nonrigid cusp is the Picard orbifold (the quotient of the hyperbolic three-dimensional space by the Picard group) which is covered by the complement of the Borromean rings; and the orbifolds obtainable by surgery on the Picard orbifold are covered by manifolds (generally, by cone-manifolds) obtainable by suitable surgeries on the Borromean rings. In this paper we establish an exact correspondence between the surgery parameters on the Adams orbifolds and their covering manifolds. This makes it possible to use the computer program SnapPea for calculating the volumes of hyperbolic 3-orbifolds. We only consider orientable 3-orbifolds, using the basic facts of the orbifold theory as in [3, 5]. Like the volumes of hyperbolic 3-manifolds, the volumes of hyperbolic 3-orbifolds form a well-ordered nondiscrete subset of order type ωω of the real axis R, and each volume is realized only for finitely many orbifolds [3]. In particular, there is a hyperbolic orbifold of smallest volume (which is not known yet), as well as the smallest limit volume. The singular set of the smallest known hyperbolic 3-orbifold is shown in Fig. 0.1 (the underlying space is the three-dimensional sphere; the edge labels 2 are omitted in the figure). Its volume is (approximately) 0.039050.

Abstract: A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated to one of the families. It is conjectured that there are exactly 6 exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea's census of closed hyperbolic 3-manifolds, associated to 5 other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated to the seventh family we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.