Heegaard splittings, the virtually Haken conjecture and Property tau
Marc Lackenby
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I will outline the interaction of three
seemingly disparate topics: Heegaard splittings,
the virtually Haken conjecture and Property
tau. The latter is concept due to Lubotzky and
Zimmer, that is defined in terms of eigenvalues
of the Laplacian, graph theory or representation
theory, and is related to Property T. I will
formulate a conjecture about Heegaard splittings,
and will show how this and a conjecture of
Lubotzky and Sarnak about Property tau implies
the virtually Haken conjecture for hyperbolic
3-manifolds. I will also show that the positive
virtual $b_1$ conjecture has equivalent
formulations in terms of Heegaard splittings,
and in terms of the behavior of the Laplacian.
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