  We examine three key conjectures in 3-manifold theory: the virtually Haken
conjecture, the positive virtual b_1 conjecture and the virtually fibred
conjecture. We explore the interaction of these conjectures with the following
seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard
splittings.
  We first give a necessary and sufficient condition, in terms of spectral
geometry, for a finitely presented group to have a finite index subgroup with
infinite abelianisation. For negatively curved 3-manifolds, we show that this
is equivalent to a statement about generalised Heegaard splittings. We also
formulate a conjecture about the behaviour of Heegaard genus under finite
covers which, together with a conjecture of Lubotzky and Sarnak about Property
tau, would imply the virtually Haken conjecture for hyperbolic 3-manifolds.
Along the way, we prove a number of unexpected theorems about 3-manifolds. For
example, we show that for any closed 3-manifold that fibres over the circle
with pseudo-Anosov monodromy, any cyclic cover dual to the fibre of
sufficiently large degree has an irreducible, weakly reducible Heegaard
splitting. Also, we establish lower and upper bounds on the Heegaard genus of
the congruence covers of an arithmetic hyperbolic 3-manifold, which are linear
in volume.
