The asymptotic behaviour of Heegaard genus

TL;DR

This paper investigates the growth rates of Heegaard genus in finite covers of negatively curved 3-manifolds, providing evidence for conjectures relating genus growth to topological properties like fibering and Betti numbers.

Contribution

It establishes bounds on the growth of Heegaard genus in covers and links this growth to the manifold's fibering and Betti number properties, supporting key conjectures.

Findings

Heegaard genus growth slower than square root implies positive first Betti number.

Strong Heegaard genus cannot grow slower than square root of degree.

Slower than fourth root growth implies the manifold fibers over the circle.

Abstract

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of M_i cannot grow more slowly than the square root of d_i. (3) If the Heegaard genus of M_i grows more slowly than the fourth root of d_i, then M_i 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.