Geometry & Topology

The virtual Haken conjecture: Experiments and examples

Nathan M Dunfield and William P Thurston

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A 3–manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3–manifold with infinite fundamental group has a finite 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 Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3–manifolds, and for each of them found finite 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 filling. In particular, we show that if a 3–manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.

Article information

Geom. Topol., Volume 7, Number 1 (2003), 399-441.

Received: 30 September 2002
Accepted: 13 April 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus 57M10: Covering spaces
Secondary: 57M27: Invariants of knots and 3-manifolds 20E26: Residual properties and generalizations; residually finite groups 20F05: Generators, relations, and presentations

virtual Haken Conjecture experimental evidence Dehn filling one-relator quotients figure-8 knot


Dunfield, Nathan M; Thurston, William P. The virtual Haken conjecture: Experiments and examples. Geom. Topol. 7 (2003), no. 1, 399--441. doi:10.2140/gt.2003.7.399.

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