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Published June 24, 2003 | public
Journal Article Open

The virtual Haken conjecture: Experiments and examples


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.

Additional Information

© Geometry & Topology Publications Submitted to GT on 30 September 2002. Paper accepted 13 April 2003. Paper published 24 June 2003. The first author was partially supported by an NSF Postdoctoral Fellowship. The second author was partially supported by NSF grants DMS-9704135 and DMS-0072540. We would like to thank Ian Agol, Daniel Allcock, Matt Baker, Danny Calegari, Greg Kuperberg, Darren Long, Alex Lubotzky, Alan Reid, William Stein, and Dylan Thurston for useful conversations. We also thank of the authors of the computer programs SnapPea [56] and GAP [28] which were critical for our computations.


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