Related papers: Geometric triangulations and highly twisted links

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher,…

We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1.…

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In…

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete…

We prove that the knots and links in the infinite set of $3$-highly twisted $2m$-plats, with $m \geq 2$, are all hyperbolic. This should be compared with a result of Futer-Purcell for $6$-highly twisted diagrams. While their proof uses…

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of…

According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most \pi can be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local…

We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.

We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this…

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many…

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings;…

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a…

Myers shows that every compact, connected, orientable $3$--manifold with no $2$--sphere boundary components contains a hyperbolic knot. We use work of Ikeda with an observation of Adams-Reid to show that every $3$--manifold subject to the…

In this paper we construct two different explicit triangulations of the family of double twist knots $K(p,q)$ using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the…

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.

Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits…

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We…

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become…