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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57M05, 57M10

Secondary: 57M27, 20E26, 20F05

References

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Publication

Received: 30 September 2002

Accepted: 13 April 2003

Published: 24 June 2003

Proposed: Jean-Pierre Otal

Seconded: Walter Neumann, Martin Bridson

Authors

Nathan M Dunfield
Department of Mathematics Harvard University Cambridge Massachusetts 02138 USA

William P Thurston
Department of Mathematics University of California at Davis Davis California 95616 USA

Volume 7, issue 1 (2003)

Nathan M Dunfield and William P Thurston