Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

It is well known that for a closed hyperbolic 3-manifold $M$ the rank of $\pi_1(M)$ is bounded above by some universal constant $K$ times the volume of $M$. Using similar methods, i.e. the thick-thin decomposition of $M$, one can also show that the Heegaard genus of $M$ is bounded above by a universal constant times the volume of $M$. (I believe that Thurston showed this first, though I am not sure as to how.)

I am looking to construct a sequence of (closed) hyperbolic 3-manifolds, say {$M_n$} such that the volume grows linearly in the Heegaard genus of $M_n$. That is, I am trying to show that a linear bounded on Heegaard genus in terms of volume is the best one can do. So far, I am having some trouble constructing such an example.

Does anyone have a good method or reference for constructing such an example? Also, another approach to the problem would be wonderful (short of solving the rank versus Heegaard genus conjecture of hyperbolic 3-manifolds, of course).

share|improve this question
9  
Take a 3-manifold group that maps onto a free group, and take induced covers (the rank and volume will both grow linearly). See also: ams.org/mathscinet-getitem?mr=2218779 front.math.ucdavis.edu/0709.0101 –  Ian Agol Apr 11 '10 at 5:53

1 Answer 1

up vote 2 down vote accepted

Agol says: Take a 3-manifold group that maps onto a free group, and take induced covers (the rank and volume will both grow linearly). See this paper of Lackenby.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.