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The virtual Haken conjecture: Experiments and examples

Nathan M Dunfield and William P Thurston

Geometry & Topology 7 (2003) 399–441

arXiv: math.GT/0209214


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.

virtual Haken Conjecture, experimental evidence, Dehn filling, one-relator quotients, figure-8 knot
Mathematical Subject Classification 2000
Primary: 57M05, 57M10
Secondary: 57M27, 20E26, 20F05
Forward citations
Received: 30 September 2002
Accepted: 13 April 2003
Published: 24 June 2003
Proposed: Jean-Pierre Otal
Seconded: Walter Neumann, Martin Bridson
Nathan M Dunfield
Department of Mathematics
Harvard University
Massachusetts 02138
William P Thurston
Department of Mathematics
University of California at Davis
California 95616