234 research outputs found
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
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
Triangulations of 3-manifolds with essential edges
We define essential and strongly essential triangulations of 3-manifolds, and
give four constructions using different tools (Heegaard splittings, hierarchies
of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian
manifolds) to obtain triangulations with these properties under various
hypotheses on the topology or geometry of the manifold. We also show that a
semi-angle structure is a sufficient condition for a triangulation of a
3-manifold to be essential, and a strict angle structure is a sufficient
condition for a triangulation to be strongly essential. Moreover, algorithms to
test whether a triangulation of a 3-manifold is essential or strongly essential
are given.Comment: 30 pages, 14 figures. Exposition improve
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