Craig D. Hodgson

Spherical space forms and Dehn filling

Abstract This paper concerns those Dehn fillings on a torally bounded 3-manifold which yield manifolds with a finite fundamental group. The focus will be on those torally bounded 3-manifolds which either contain an essential torus, or whose interior admits a complete hyperbolic structure. While we give several general results, our sharpest theorems concern Dehn fillings on manifolds which contain an essential torus. One of these results is a sharp “finite surgery theorem.” The proof incl udes a characterization of the finite fillings on “generalized” iterated torus knots with a complete classification for the iterated torus knots in the 3-sphere. We also give a proof of the so-called “2π” theorem of Gromov and Thurston, and obtain an improvement (by a factor of two) in the original estimates of Thurston on the number of non-negatively-curved Dehn fillings on a torally bounded 3-manifold whose interior admits a complete hyperbolic structure.

Symmetries, isometries and length spectra of closed hyperbolic three-manifolds

Previously known algorithms to compute the symmetry group of a cusped hyperbolic three-manifold and to test whether two cusped hyperbolic three-manifolds are isometric do not apply directly to closed manifolds. But by drilling out geodesics from closed manifolds one may compute their symmetry groups and test for isometries using the cusped manifold techniques. To do so, one must know precisely how many geodesics of a given length the closed manifold has. Here we prove the propositions needed to rigorously compute a length spectrum, with multiplicities. We also tabulate the symmetry groups of the smallest known closed hyperbolic three-manifolds.

Computing arithmetic invariants of 3-manifolds

Snap is a computer program for computing arithmetic invariants of hyperbolic 3-manifolds, built on Jeff Weekss SnapPea and the number theory package Pari. Its approach is to compute the hyperbolic structure to very high prec ision, and use th is to find an exact description of the structure. Then the correctness of the hyperbolic structure can be verified, and the arithmetic invariants of Neumann and Reid can be computed. Snap also computes high precision numerical invariants such as volume, Chern–Simons invariant, eta invariant, and the Borel regulator.

A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere

We describe a characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in E 3 all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832

Commensurators of Cusped Hyperbolic Manifolds

This paper describes a general algorithm for finding the commensurator of a nonarithmetic hyperbolic manifold with cusps and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all nonarithmetic hyperbolic once-punctured torus bundles over the circle. For hyperbolic 3-manifolds, the algorithm has been implemented using Goodmans computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most seven ideal tetrahedra, and for the complements of hyperbolic knots and links with up to twelve crossings.

1-efficient triangulations and the index of a cusped hyperbolic 3-manifold

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3‐manifold M (a collection of q ‐series with integer coefficients, introduced by Dimofte, Gaiotto and Gukov) to a topological invariant of oriented cusped hyperbolic 3‐manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1‐efficient and (b) if M is hyperbolic, it has a canonical set of 1‐efficient ideal triangulations related by 2‐3 and 0‐2 moves which preserve the 3D index. We illustrate our results with several examples. 57N10, 57M50; 57M25

Veering triangulations admit strict angle structures

Agol recently introduced the concept of a veering taut triangulation of a 3–manifold, which is a taut ideal triangulation with some extra combinatorial structure. We define the weaker notion of a “veering triangulation” and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, giving an example of a veering taut triangulation that is not layered.

Eigenvalue estimates and isoperimetric inequalities for cone-manifolds

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others. We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates. We also establish a version of the Levy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

Non-geometric Veering Triangulations

Recently, Ian Agol introduced a class of “veering” ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are “geometric”, i.e. realized in the complete hyperbolic metric with all tetrahedra positively oriented. This article describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra.