We examine three key conjectures in 3-manifold theory: the virtually Hakenconjecture, the positive virtual b_1 conjecture and the virtually fibredconjecture. We explore the interaction of these conjectures with the followingseemingly unrelated areas: eigenvalues of the Laplacian, and Heegaardsplittings. We first give a necessary and sufficient condition, in terms of spectralgeometry, for a finitely presented group to have a finite index subgroup withinfinite abelianisation. For negatively curved 3-manifolds, we show that thisis equivalent to a statement about generalised Heegaard splittings. We alsoformulate a conjecture about the behaviour of Heegaard genus under finitecovers which, together with a conjecture of Lubotzky and Sarnak about Propertytau, would imply the virtually Haken conjecture for hyperbolic 3-manifolds.Along the way, we prove a number of unexpected theorems about 3-manifolds. Forexample, we show that for any closed 3-manifold that fibres over the circlewith pseudo-Anosov monodromy, any cyclic cover dual to the fibre ofsufficiently large degree has an irreducible, weakly reducible Heegaardsplitting. Also, we establish lower and upper bounds on the Heegaard genus ofthe congruence covers of an arithmetic hyperbolic 3-manifold, which are linearin volume.
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