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.



Heegaard splitting of covering hyperbolic manifold.
There are examples like this. Check out section 4.5 of Shalen's paper "Hyperbolic volume, Heegaard genus and ranks of groups." It's here: <http://arxiv.org/abs/0904.0191>


He gives a reference for a genus 3 example by Alan Reid and a sketch of a technique for producing examples by Hyam Rubinstein. Shalen also conjectures that the genus can drop by at most 1 in a finite cover of a closed hyperbolic 3-manifold.


