Related papers: Angle structure on general hyperbolic 3-manifolds

We give a bounded runtime solution to the homeomorphism problem for closed hyperbolic 3-manifolds. This is an algorithm which, given two triangulations of hyperbolic 3-manifolds by at most $t$ tetrahedra, decides if they represent the same…

Cannon and Swenson have shown that each hyperbolic 3-manifold group has a natural subdivision rule on the space at infinity, and that this subdivision rule captures the action of the group on the sphere. Explicit subdivision rules have also…

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic…

A divide is the image of a proper and generic immersion of a compact $1$-manifold into the $2$-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. In this paper, we reveal a hidden hyperbolic structure in…

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles…

In this paper, we show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.

Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering…

We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable 3-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been…

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint…

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework…

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…

The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal…

In this paper, we give a method to construct holonomy matrices of hyperbolic 3-manifolds by extending the known method of hyperbolic 2-manifolds. It enables us to consider hyperbolic 3-manifolds with nontrivial holonomies. We apply our…

We determine the combinatorial types of all the 3-dimensional simple convex polytopes in R^3 that can be realized as mean curvature convex (or totally geodesic) Riemannian polyhedra with non-obtuse dihedral angles in Riemannian 3-manifolds…

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an…

Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by…

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges…

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We…

The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic…