Geometry & Topology, Vol. 7 (2003) Paper no. 12, pages 399--441.

The virtual Haken conjecture: Experiments and examples

Nathan M Dunfield, William P Thurston

Abstract. 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.

Keywords. Virtual Haken Conjecture, experimental evidence, Dehn filling, one-relator quotients, figure-8 knot

AMS subject classification. Primary: 57M05, 57M10. Secondary: 57M27, 20E26, 20F05.

DOI: 10.2140/gt.2003.7.399

E-print: arXiv:math.GT/0209214

Submitted to GT on 30 September 2002. Paper accepted 13 April 2003. Paper published 24 June 2003.

Notes on file formats

Nathan M Dunfield, William P Thurston
Department of Mathematics, Harvard University
Cambridge MA, 02138, USA
Department of Mathematics, University of California, Davis
Davis, CA 95616, USA


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