The virtual Haken conjecture: Experiments and examples

By Nathan M. Dunfield, William P. Thurston. 2002

85 citations recorded by the source corpus.

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-manifolds, and for each of them found nite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn lling. In particular, we show that if a 3-manifold with torus boundary has a Seifert bered Dehn lling with hyperbolic base orbifold, then most of the Dehn lled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the gure-8 knot is virtually Haken. AMS Classication numbers Primary: 57M05, 57M10 Secondary: 57M27, 20E26, 20F05

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/93364154a0f0da03842fe8b6e5d60746c9bd59b6

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Citations in the corpus

Finite covers of random 3-manifolds (2005, methodology): “ageable size, and a 2-generator group has about 1.3% probability. To test our thinking, we made 1000 random 2-generator presentations with statistics similar to the census manifolds used in our paper [DT2], and computed all epimorphisms to the ﬁrst few non-abelian simple groups. Only 15 of these gro”
Heegaard splittings, the virtually Haken conjecture and Property (τ) (2002, background): “ﬁnite cover with positive ﬁrst Betti number. A recent analysis by Dunﬁeld and Thurston of the 10986 closed hyperbolic 3- manifolds in theHodgson-Weekscensusshowed thatthey all satisfy this conjecture [13]. The following conjecture, due to Thurston [58], is stronger still. Virtually ﬁbred conjecture.”
THE DEFORMATION SPACES OF CONVEX RP²-STRUCTURES ON 2-ORBIFOLDS (2001, background): “A subgroup Γ of GL(3,R) is positive proximal if all proximal elements of Γ are positive proximal.”
Orderable 3-manifold groups (2002, background): “−4 <p q <4, π1(MK( p q )) admits a representation to PSL2(R) with non-abelian image and hence is virtually left-orderable (when p q = 0 apply Corollary 3.4). We remark that Dunﬁeld and Thurston [DT] have proven that each MK( p q ), p q 6= ∞, has a ﬁnite cover with a positive ﬁrst Betti number ”
The virtual Haken conjecture (2012, background): “Kirby’s problem list, as well as [43, Questions 15-18]. There has been much work on the virtual Haken conjecture before for certain classes of manifolds. These include manifolds in the Snappea census [14], surgeries on various classes of cusped hyperbolic manifolds [2, 3, 4, 7, 10, 11, 12, 27, 33, 3”