TITLE: Heegaard splitting of covering hyperbolic manifold.
QUESTION [13 upvotes]: I am curious about how the Heegaard genus changes after a finite covering. 
Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that 
the Heegaard genus of a finite covering of $N$ is less than the Heegaard genus of $N$? 
Thank you!
Note: Heegaard genus of a 3-manifold means the minimal genus of all Heegaard splittings.

REPLY [6 votes]: Hyam Rubinstein and me have results about the behavior of the Heegaard genus under double covers for non-Haken manifolds, see http://arxiv.org/abs/math/0607145.  Essentially, we show that the Heegaard genera of the two manifolds bound each other linearly.  (The statement is a little more complicated for branched covers.)