Related papers: Toroidal and annular Dehn fillings

For a closed orientable connected 3-manifold $M$, its complexity $\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its triangulations. Under the assumption that $M$ is prime (but not necessarily atoroidal), we…

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…

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that…

We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants…

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show…

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These…

In this paper, we study complete $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $\delta$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $\delta>\delta_0(n)$,…

Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In…

Let $M$ be a 1-cusped hyperbolic 3-manifold. In this paper, we study the behavior of $N_M(v)$, the number of Dehn fillings of $M$ with a given volume $v(\in \mathbb{R})$. We conduct extensive computational experiments to estimate $N_M$ and…

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy:…

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally…

Let M be a closed orientable irreducible 3-manifold, and let f be a diffeomorphism over M. We call an embedded 2-torus T an Anosov torus if it is invariant and the induced action of f over \pi_1(T) is hyperbolic. We prove that only few…

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an…

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a…

In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$…

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the…

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal…

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.

Classical work of Thurston and Gabai shows that finitely many taut sutured manifold hierarchies determine the Thurston norm of a compact oriented irreducible $3$-manifold with toroidal boundary. We give an explicit procedure to extract this…

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and…