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.