The shape of hyperbolic Dehn surgery space

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848 around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (e.g. volumes and core geodesic lengths) during hyperbolic Dehn filling. This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups.Comment: 46 pages, 3 figure

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. realised in the complete hyperbolic metric with all tetrahedra positively oriented. This paper 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

Veering triangulations admit strict angle structures

Agol recently introduced the concept of a veering taut triangulation, which is a taut 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.Comment: 15 pages, 9 figure

Quadrilateral-octagon coordinates for almost normal surfaces

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using cohomology, plus other minor changes; v3: Minor housekeepin

Mid-infrared InAs/InAsSb superlattice nBn photodetector monolithically integrated onto silicon

Mid-infrared (MIR) silicon photonics holds the potential for realizing next generation ultracompact spectroscopic systems for applications in gas sensing, defense, and medical diagnostics. The direct epitaxial growth of antimonide-based compound semiconductors on silicon provides a promising approach for extending the wavelength of silicon photonics to the longer infrared range. This paper reports on the fabrication of a high performance MIR photodetector directly grown onto silicon by molecular beam epitaxy. The device exhibited an extended cutoff wavelength at ∼5.5 μm and a dark current density of 1.4 × 10–2 A/cm2 under 100 mV reverse bias at 200 K. A responsivity of 0.88 A/W and a specific detectivity in the order of 1.5 × 1010 Jones was measured at 200 K under 100 mV reverse bias operation. These results were achieved through the development of an innovative structure which incorporates a type-II InAs/InAsSb superlattice-based barrier nBn photodetector grown on a GaSb-on-silicon buffer layer. The difficulties in growing GaSb directly on silicon were overcome using a novel growth procedure consisting of an efficient AlSb interfacial misfit array, a two-step growth temperature procedure and dislocation filters resulting in a low defect density, antiphase domain free GaSb epitaxial layer on silicon. This work demonstrates that complex superlattice-based MIR photodetectors can be directly integrated onto a Si platform, which provides a pathway toward the realization of new, high performance, large area focal plane arrays and mid-infrared integrated photonic circuits

Triangulations of hyperbolic 3-manifolds admitting strict angle structures

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis, and an example illustrating that the triangulations produced using our methods are not generally geometric.Comment: 28 pages, 9 figures. Minor edits and clarification based on referee's comments. Corrected proof of Lemma 7.4. To appear in the Journal of Topolog