The virtual Haken conjecture: Experiments and examples

  title={The virtual Haken conjecture: Experiments and examples},
  author={Nathan M. Dunfield and William P. Thurston},
  journal={Geometry \& Topology},
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with innite fundamental group has a nite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete HodgsonWeeks census of 10,986 small-volume closed hyperbolic 3… 
Finite covers of random 3-manifolds
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which
Finite covering spaces of 3-manifolds
Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understand- ing of closed hyperbolic 3-manifolds
Automorphic forms and rational homology 3--spheres
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
Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston
We will discuss the proof of Waldhausen’s conjecture that compact aspherical 3-manifolds are virtually Haken, as well as Thurston’s conjecture that hyperbolic 3manifolds are virtually fibered. The
Covering Spaces of Arithmetic 3-Orbifolds
Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index
The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are
Heegaard splittings, the virtually Haken conjecture and Property (τ)
The behaviour of finite covers of 3-manifolds is a very important, but poorly understood, topic. There are three, increasingly strong, conjectures in the field that have remained open for over twenty
This short note is a follow-up to (the second part of) Section 7.6 of my book [3]. There, earlier results of N. Dunfield and W. Thurston [1] on a certain type of random 3-manifolds (compact,
A subgroup of a group is known as a surface subgroup if it is isomorphic to the fundamental group of a closed connected orientable surface with positive genus. The surface subgroup conjecture in
Poincaré conjecture and related statements
The main topics of this paper are mathematical statements, results or problems related with the Poincare conjecture, a recipe to recognize the three-dimensional sphere. The statements, results and


Virtual Haken 3-manifolds and Dehn " lling
A stronger conjecture is that any closed, connected, orientable, irreducible 3-manifold= with in"nite fundamental group has a virtually positive "rst Betti number, i.e.= has a "nite cover which has
Virtually Haken Dehn-lling
We show that \most" Dehn-llings of a non-bered, atoroidal, Haken three-manifold with torus boundary are virtually Haken. 1 Results Suppose that X is a compact, oriented, three-manifold with boundary
Heegaard splittings of branched coverings of
This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other
Alexander and Thurston norms of fibered 3-manifolds
For a 3-manifold M, McMullen derived from the Alexander polynomial of M a norm on H^1(M, R) called the Alexander norm. He showed that the Thurston norm on H^1(M, R), which measures the complexity of
Bounded, separating, incompressible surfaces in knot manifolds
This generalizes and strengthens the main theorem of [13]. Note that the hypothesis of Theorem 1 is satisfied whenever M is a knot manifold, i.e. the complement of an open tubular neighborhood of a
Free quotients and the first betti number of some hyperbolic manifolds
In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic
Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some
Problems in low-dimensional topology
Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of