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
and
Department of Mathematics, University of California,
Davis
Davis, CA 95616, USA
Email:
nathand@math.harvard.edu, wpt@math.ucdavis.edu
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