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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.

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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.

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  • $\begingroup$ I think there's a variation on Hyam's construction, where you can take a non-orientable manifold with a non-orientable Heegaard splitting, whose 2-fold orientation cover has an orientable splitting of smaller genus than a Heegaard splitting downstairs. $\endgroup$
    – Ian Agol
    Commented Jan 27, 2012 at 21:46
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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.)

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