  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.
