The asymptotic behaviour of Heegaard genus

الأصل · EN

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let Mᵢ be a collection of finite regular covers with degree dᵢ. (1) If the Heegaard genus of Mᵢ grows more slowly than the square root of dᵢ, then Mᵢ has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of Mᵢ cannot grow more slowly than the square root of dᵢ. (3) If the Heegaard genus of Mᵢ grows more slowly than the fourth root of dᵢ, then Mᵢ fibres over the circle for all sufficiently large i. These results provide supporting evidence for the Heegaard gradient conjecture and the strong Heegaard gradient conjecture. As a corollary to (3), we give a necessary and sufficient condition for M to be virtually fibred in terms of the Heegaard genus of its finite covers.