Heegaard splittings, the virtually Haken conjecture and Property (τ)

The behaviour of finite covers of 3-manifolds is a very important, but poorly understood, topic. There are three, increasingly strong, conjectures in the field that have remained open for over twenty years – the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture. Any of these would have profound ramifications for 3-manifold theory. In this paper, we explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings.

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